Solution:
We divide the path into eight “$R$” movements and eight “$U$” movements. Five sections of alternative $RURUR$ or $URURU$ are necessary in order to make four “turns.” We use the first case and multiply by $2$.

For $U$, we have seven ordered pairs of positive integers $(a,b)$ such that $a+b=8$.
For $R$, we subtract $1$ from each section (to make the minimum stars of each section $0$) and we use Stars and Bars to get ${7 \choose 5}=21$.

Thus our answer is $7\cdot21\cdot2=\boxed{294}$.
~eevee9406
Draw a few examples of the path. However, notice one thing in common - if the path starts going up, there will be 3 "segments" where the path goes up, and two horizontal "segments." Similarly, if the path starts going horizontally, we will have three horizontal segments and two vertical segments. Those two cases are symmetric, so we only need to consider one. If our path starts going up, by stars and bars, we can have $\binom{7}{2}$ ways to split the 8 up's into 3 lengths, and have $\binom{7}{1}$ to split the 8-horizontals into 2 lengths. We multiply them together, and multiply by 2 for symmetry, giving us $2*\binom{7}{2}*\binom{7}{1}=294$.
~nathan27 (original by alexanderruan)
Notice that the $RURUR$ case and the $URURU$ case is symmetrical. WLOG, let's consider the RURUR case.
Now notice that there is a one-to-one correspondence between this problem and the number of ways to distribute 8 balls into 3 boxes and also 8 other balls into 2 other boxes, such that each box has a nonzero amount of balls.
There are ${8+2-3 \choose 2}$ ways for the first part, and ${8+1-2 \choose 1}$ ways for the second part, by stars and bars.
The answer is $2\cdot {7 \choose 2} \cdot {7 \choose 1} = \boxed{294}$.
~northstar47
Feel free to edit this solution
Starting at the origin, you can either first go up or to the right. If you go up first, you will end on the side opposite to it (the right side) and if you go right first, you will end up on the top. It can then be observed that if you choose the turning points in the middle $7 \times 7$ grid, that will automatically determine your start and ending points. For example, in the diagram if you choose the point $(3,2)$ and $(5,3)$, you must first move three up or two right, determining your first point, and move 5 up or 3 right, determining your final point. Knowing this is helpful because if we first move anywhere horizontally, we have $7$ points on each column to choose from and starting from left to right, we have $6,5,4,3,2,1$ points on that row to choose from. This gives us $7(6)+7(5)+7(4)+7(3)+7(2)+7(1)$ which simplifies to $7\cdot21$. The vertical case is symmetrical so we have $7\cdot21\cdot2 = \boxed{294}$
~KEVIN_LIU
As in Solution 1, there are two cases: $RURUR$ or $URURU$. We will work with the first case and multiply by $2$ at the end. We use stars and bars; we can treat the $R$s as the stars and the $U$s as the bars. However, we must also use stars and bars on the $U$s to see how many different patterns of bars we can create for the reds. We must have $1$ bar in $8$ blacks, so we use stars and bars on the equation \[x + y = 8\]. However, each divider must have at least one black in it, so we do the change of variable $x' = x-1$ and $y' = x-1$. Our equation becomes \[x' + y' = 6\]. By stars and bars, this equation has $\binom{6 + 2 - 1}{1} = 7$ valid solutions. Now, we use stars and bars on the reds. We must distribute two bars amongst the reds, so we apply stars and bars to \[x + y + z = 8\]. Since each group must have one red, we again do a change of variables with $x' = x-1$, $y' = y-1$, and $z' = z-1$. We are now working on the equation \[x' + y' + z' = 5\]. By stars and bars, this has $\binom{5 + 3 - 1}{2} = 21$ solutions. The number of valid paths in this case is the number of ways to create the bars times the number of valid arrangements of the stars given fixed bars, which equals $21 \cdot 7 = 147$. We must multiply by two to account for both cases, so our final answer is $147 \cdot 2 = \boxed{294}$.
~ [cxsmi](https://artofproblemsolving.comhttps://artofproblemsolving.com/wiki/index.php/User:Cxsmi)

Answer: 294

<｜begin▁of▁sentence｜><｜User｜>Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.
[asy] size(10cm); usepackage("tikz");label("egin{tikzpicture}[scale=.5]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\end{tikzpicture}",origin); label("\begin{tikzpicture}[scale=.5]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\end{tikzpicture}",E); [/asy]<｜Assistant｜><think>

Okay, so I need to find the number of paths from the lower left corner to the upper right corner of an 8x8 grid, moving only along the lines, with a path length of 16. The key here is that the path must change direction exactly four times. Hmm, let me think through this step by step.

First, let me recall that on an 8x8 grid, moving from the bottom left (0,0) to the top right (8,8), each path consists of a series of moves either right or up. Since the grid is 8x8, you need to move 8 steps to the right and 8 steps up, totaling 16 steps. So any path from start to finish is a permutation of these moves. The total number of such paths without any restrictions is C(16,8), which is 12870. But here, we have a restriction: the path must change direction exactly four times.

Wait, direction changes. Let me clarify what a direction change means here. Each time the path switches from moving right to up or from up to right, that's a direction change. So, for example, if you go right for a few steps, then up for a few steps, that's one direction change. Each time you switch direction, that's another change. The problem states exactly four direction changes.

Let me think of how direction changes relate to the structure of the path. A path that changes direction four times would have five segments, right? Because each direction change starts a new segment. For instance, if you start going right, then up, that's one change, making two segments. So four changes would mean five segments. Each segment is a consecutive sequence of right or up moves.

Since the path starts at (0,0) and ends at (8,8), the first move can be either right or up. Let me consider both possibilities, but maybe it's symmetric? Let me see.

Suppose the path starts with a right move. Then the segments would alternate between right and up. Since there are five segments, the directions would go right, up, right, up, right. Wait, but five segments with four direction changes would alternate starting with right or up. Similarly, if it starts with up, then the directions would be up, right, up, right, up. However, since we need to end at (8,8), the last segment must be in the direction of the final coordinate. Wait, actually, the last segment could be either right or up, depending on the path. Hmm, maybe I need to think more carefully.

Alternatively, perhaps the number of direction changes is independent of starting direction. Let me formalize this.

Let me denote the direction changes as follows. Each time the direction changes, it switches between horizontal (right) and vertical (up). So a path with four direction changes will have five segments. The first segment can be either right or up, then alternates. Since the total number of right moves must be 8 and up moves must be 8, the number of segments in each direction must satisfy certain conditions.

Suppose the path starts with a right segment. Then the segments alternate between right and up. Since there are five segments, starting with right would mean the segments are: right, up, right, up, right. That would be three right segments and two up segments. But the total right moves would then be the sum of the lengths of the right segments, and similarly for up. However, the total right moves need to be 8, and up moves also 8. So in this case, three right segments and two up segments. Let me denote the lengths of the right segments as r1, r2, r3 and the up segments as u1, u2. Then:

r1 + r2 + r3 = 8

u1 + u2 = 8

Similarly, if the path starts with an up segment, then the segments would be up, right, up, right, up. So three up segments and two right segments. Then:

u1 + u2 + u3 = 8

r1 + r2 = 8

Therefore, depending on whether the first move is right or up, the number of segments in each direction differs. Since the problem doesn't specify the starting direction, we need to consider both cases and add them together.

Therefore, the total number of paths with exactly four direction changes is equal to the number of paths starting with right (with three right segments and two up segments) plus the number starting with up (three up segments and two right segments).

So, let me compute both cases.

First case: starting with right.

We have three right segments and two up segments. The direction sequence is R, U, R, U, R. The total right steps: r1 + r2 + r3 =8, and up steps: u1 + u2 =8. Each of these segments must have at least 1 step, because you can't have a segment of length 0 (since that would mean an immediate direction change without moving). Wait, actually, in grid paths, each segment must have at least length 1, otherwise, you can't have a direction change without moving. So each segment length must be at least 1.

Therefore, for the right segments: r1, r2, r3 ≥1, and similarly for up segments u1, u2 ≥1.

Therefore, the number of solutions for r1 + r2 + r3 =8 with each ri ≥1 is C(8-1,3-1)=C(7,2)=21. Similarly, for u1 + u2 =8 with each ui ≥1, it's C(8-1,2-1)=C(7,1)=7. Therefore, the number of paths in this case is 21 *7 = 147. But wait, is that all?

Wait, but actually, the order of the segments is fixed by the direction sequence. Since the direction alternates, the segments are in a specific order. So once we have the lengths of the right and up segments, the path is determined. Therefore, the total number for the first case (starting with right) is indeed 21 *7 = 147.

Similarly, for the second case, starting with up. Then we have three up segments and two right segments. The direction sequence is U, R, U, R, U. The up segments: u1 + u2 + u3 =8, each ≥1, so the number of solutions is C(8-1,3-1)=C(7,2)=21. The right segments: r1 + r2 =8, each ≥1, so C(7,1)=7. Therefore, the number here is also 21 *7 =147.

Therefore, total paths would be 147 +147 = 294. Wait, but hold on, is that correct?

Wait a second, maybe I need to check if the direction changes are exactly four. Let me confirm. Each direction change is a switch between right and up. So starting with right, the direction changes occur after the first right segment (to up), then after the first up segment (to right), then after the second right segment (to up), then after the second up segment (to right). So that's four direction changes, resulting in five segments. Similarly for starting with up. So that seems correct.

But let me think again. Let me take a small example. Suppose a 2x2 grid, going from (0,0) to (2,2). Let's see how many paths with exactly two direction changes. Following the same logic, starting with right: segments R, U, R. So two direction changes? Wait, no. Wait, starting with R, then U is first change, then R is second change. Wait, actually, the number of direction changes is one less than the number of segments. Wait, hold on, maybe my initial reasoning was wrong.

Wait, if you have five segments, then the number of direction changes is four, because each segment after the first requires a direction change. So five segments mean four direction changes. So that part is correct. So in the 2x2 grid example, if we want paths with two direction changes, that would be three segments. Let's see. For a 2x2 grid, moving from (0,0) to (2,2). A path with two direction changes would have three segments. Starting with right: R, U, R. The total right steps: r1 + r2 =2, with r1, r2 ≥1. So r1=1, r2=1. Similarly, the up steps: u1 must be 2? Wait, no. Wait, the up segment is between the two right segments. So the up segment length must be u1, which has to be at least 1. So total up steps: u1 =2? Wait, no, in this case, the up segment is only one segment. Wait, starting with R, then U, then R. The up segment is in the middle. So the up steps total to u1, which must be 2? Wait, but in a 2x2 grid, you need two up steps. So if the path is R1, U2, R2? Wait, but the total right steps would be R1 + R2 =2, and up steps U1=2. So for example, the path could be right 1, up 2, right 1. That would be a valid path with two direction changes. Alternatively, right 2, up 1, right 0? Wait no, segments must be at least 1. Wait, so in this case, the number of paths starting with R would be the number of solutions to R1 + R2 =2 with R1,R2 ≥1, which is 1 (since 1+1=2). Similarly, the up segment must be exactly 2. Wait, but the up segment can be split? Wait no, in this case, starting with R, the up segment is only one segment, so U1 must be exactly 2. Wait, but in the path, after the first R segment, you go up, then right again. So the up segment has to cover all the remaining up steps? Wait, no, in the 2x2 grid, if you start with R1, then U2 would take you to (1,2), then R1 to (2,2). That works. Alternatively, starting with R2, then U2, but that would be only one direction change. Wait, R2 then U2 would be two segments, so one direction change. So to have two direction changes, you need three segments. So R1, U2, R1 is two direction changes. Similarly, R1, U1, R1, U1 would be three direction changes, but that would be four segments. Wait, maybe my small example is getting confusing.

Alternatively, perhaps my initial calculation is correct, but let me check with the small grid. Let me compute the number of paths with exactly two direction changes on a 2x2 grid. The total paths are C(4,2)=6. The paths with two direction changes would be those that have three segments. Let's list them:

Starting with Right:

1. Right 1, Up 1, Right 1, Up 1: but that's four segments, three direction changes.

Wait, maybe I need to think again. Let me list all paths:

1. All rights first, then ups: R,R,U,U. This has one direction change.

2. R,U,R,U: this has three direction changes.

3. R,U,U,R: two direction changes?

Wait, let's parse each path step by step.

Wait, actually, direction changes are counted whenever the direction changes. So for the path R, U, R, U: after each step, but direction changes between R and U each time. Wait, actually, direction changes occur between segments. So if the path is R1, U1, R1, U1, that's four segments, so three direction changes.

Wait, perhaps the direction change count is the number of times the direction changes, so between segments. So for a path with n segments, there are (n-1) direction changes. Therefore, if we want exactly four direction changes, the path must have five segments. So in the original problem, five segments mean four direction changes. So in the 2x2 grid, two direction changes would mean three segments. Let me see:

Paths with three segments (two direction changes):

Starting with R:

First segment R, then U, then R. The total right steps: R1 + R2 =2, with R1, R2 ≥1. So R1=1, R2=1. The middle segment U must be U1=2. So the path is R1, U2, R1. Similarly, starting with U:

First segment U, then R, then U. The total up steps: U1 + U2 =2, so U1=1, U2=1, and the middle R segment must be R1=2. So the path is U1, R2, U1.

Therefore, there are two such paths. Alternatively, are there more?

Wait, if starting with R, could the first R segment be 2? Then the remaining R2 would be 0, which is not allowed. So no. So only one path starting with R and one starting with U, totaling two paths with two direction changes.

Alternatively, let's list all possible paths with three segments:

Starting with R:

- R1, U2, R1: this is one path.

- R2, U2: but that's two segments, one direction change.

- R1, U1, R1, U1: four segments, three direction changes.

Wait, so only one path starting with R. Similarly, starting with U:

- U1, R2, U1.

- U2, R2: one direction change.

So only one path starting with U. So total two paths with two direction changes.

Alternatively, another path starting with R: R1, U1, R1, but then you need to get to (2,2). Wait, after R1, U1, R1, you are at (2,1), so you need another U1. So that would be four segments. So indeed, only the path with R1, U2, R1 and U1, R2, U1. So two paths.

Therefore, according to my formula, for a 2x2 grid, starting with R: the right segments would be 2 segments? Wait, no. Wait in the case of starting with R, for two direction changes (three segments), the direction sequence is R, U, R. So right segments: two segments (R1 and R2), and up segments: one segment (U1). Wait, but in the formula above, for the original problem, starting with R would have three right segments if direction changes four times (five segments). Wait, in the small example, two direction changes (three segments) starting with R would have two right segments and one up segment. So the formula would be:

For the 2x2 grid, two direction changes (three segments):

Starting with R:

Number of right segments: 2 (since starting with R, alternating, so R, U, R: two R segments and one U segment). The right total is 2, so R1 + R2 =2, each ≥1: C(2-1,2-1)=C(1,1)=1. The up segment: U1=2, so only one way. So total 1*1=1.

Similarly starting with U: two up segments and one right segment. U1 + U2 =2, each ≥1: C(1,1)=1. The right segment must be R1=2, so 1 way. Total 1. So total 2, which matches.

Therefore, my formula works here. So going back to the original problem, the formula should hold.

Therefore, in the original problem, for four direction changes (five segments), starting with R would have three R segments and two U segments. The number of ways is C(8-1,3-1)*C(8-1,2-1)=C(7,2)*C(7,1)=21*7=147. Similarly starting with U, it's the same, so total 294.

Wait, but let me confirm with another small example. Let's take a 1x1 grid, going from (0,0) to (1,1). The paths are right then up, or up then right. Each has one direction change. So if we wanted paths with exactly one direction change, that's all the paths, which are two. Using the formula: For a 1x1 grid, direction changes exactly once (two segments). Starting with R: one R segment and one U segment. R1=1, U1=1. The number would be C(1-1,1-1)*C(1-1,1-1)=C(0,0)*C(0,0)=1*1=1. Similarly starting with U: 1. Total 2. Correct.

Another test: 3x3 grid, paths with three direction changes (four segments). Let's see. Starting with R: segments R, U, R, U. So two R segments and two U segments. The R total: R1 + R2 =3, each ≥1: C(3-1,2-1)=C(2,1)=2. The U total: U1 + U2 =3, each ≥1: C(2,1)=2. So total 2*2=4 starting with R. Similarly starting with U: same, so total 8. Let me see if that's correct.

In a 3x3 grid, paths with three direction changes (four segments). Starting with R:

Possible R segments: (1,2), (2,1). So R1 can be 1 or 2, then R2 is 2 or 1. The U segments similarly can be (1,2) or (2,1). So for each R split, there are two U splits, so 2*2=4. Similarly starting with U. So total 8. Let me count manually.

Starting with R:

First segment R1=1: then U segments must total 3. Suppose U1=1, then next R2=2, then U2=2. So path: R1, U1, R2, U2.

Alternatively U1=2, then U2=1: R1, U2, R2, U1.

Wait, actually, the U segments are between the R segments. So after the first R1=1, you go up U1, then right R2, then up U2. The total up must be 3: U1 + U2 =3. So if U1=1, U2=2; U1=2, U2=1; U1=3, U2=0 (invalid). So two options. Similarly for R1=2, R2=1, then U1 + U2=3, same two options. So total 2 (R splits) *2 (U splits)=4. Similarly starting with U. So total 8. That seems correct.

Therefore, my formula seems valid.

Therefore, returning to the original problem, the answer should be 294? Wait, but let me think again. Wait, in the problem statement, the examples given have five segments, right? Let me check the Asymptote code.

The first example path is (0,0) to (2,0) to (2,3) to (5,3) to (5,8) to (8,8). Let's count the direction changes here. Starting right to (2,0), then up to (2,3): that's first direction change. Then right to (5,3): second change. Then up to (5,8): third change. Then right to (8,8): fourth change. So four direction changes, five segments. Similarly the second example: (0,0) to (0,3) to (3,3) to (3,5) to (8,5) to (8,8). Starting up, then right, up, right, up. So direction changes after each segment: four changes. So that's consistent with five segments.

Therefore, the formula should hold. So 147 +147=294. But wait, hold on, the problem is on an 8x8 grid. Wait, but in the problem statement, it says "paths of length 16". Since an 8x8 grid from corner to corner requires 16 steps (8 right and 8 up), so that's correct.

But let me check if there is an oversight here. For instance, in the calculation, when starting with right, the number of right segments is three, each at least 1, so the number of ways to split 8 into three parts is C(7,2)=21, and the up segments are two parts, C(7,1)=7. So 21*7=147. Similarly for starting with up.

Wait, but is there a case where the path could start with right, but have more up segments? No, because with five segments starting with right, the sequence is R, U, R, U, R. So three R segments and two U segments. So that's correct.

Alternatively, could there be an error in assuming that all segments must be at least length 1? Yes, that's correct, because you can't have a zero-length segment; each time you change direction, you have to actually move in that direction. So each segment must be at least 1. Therefore, the stars and bars calculation with each variable at least 1 is appropriate here.

Therefore, I think the answer is 294. But let me see if that's the case. Wait, but let me check another angle.

Alternatively, maybe the problem is considering that a "direction change" is a turn, so when you go from right to up or up to right, each turn is a direction change. So four direction changes would indeed lead to five segments.

Alternatively, perhaps the formula is more general. The number of paths with exactly k direction changes is equal to 2 * C(7, k/2) * C(7, k/2) or something? Wait, no. Wait, in our case, for four direction changes, which is even? Wait, four is even, so starting with right or up would lead to different numbers of segments. Wait, four direction changes mean five segments. If starting with right, the number of right segments is (5 +1)/2 =3, and up segments 2. Similarly starting with up, up segments 3, right segments 2.

Therefore, in general, for a grid from (0,0) to (n,n), the number of paths with exactly 2m direction changes (resulting in 2m+1 segments) would be 2 * [C(2m, m) * C(2m, m)] or something? Wait, no, perhaps not exactly. Wait, in our problem, n=8, and direction changes=4, so m=2. Wait, perhaps not. Let me think again.

Alternatively, the formula for the number of paths with exactly k direction changes is 2 * C( (number of right segments -1), (something) ). Wait, perhaps the general formula is as follows:

Suppose you have an n x n grid, and you want paths from (0,0) to (n,n) with exactly k direction changes. Then, the number of such paths is 2 * C(n-1, a-1) * C(n-1, b-1), where a and b depend on the number of segments. Wait, perhaps more precisely, if the number of direction changes is k, then the number of segments is k+1. If starting with right, the number of right segments is (k+2)/2 if k is even? Wait, let's see:

If k is the number of direction changes, then the number of segments is k+1. If starting with right, then the number of right segments is ceil((k+1)/2), and up segments is floor((k+1)/2). Similarly starting with up.

Therefore, for even k: Let k=2m. Then segments =2m+1. Starting with right: right segments = m+1, up segments =m. Starting with up: up segments =m+1, right segments =m.

Therefore, for our problem, k=4 direction changes (even), so m=2. Therefore starting with right: right segments=3, up segments=2. The number of paths starting with right is C(8-1,3-1)*C(8-1,2-1)=C(7,2)*C(7,1)=21*7=147. Similarly starting with up: C(7,2)*C(7,1)=147. Total 294.

Therefore, this seems consistent.

Alternatively, perhaps the formula can be written as 2 * [C(n-1, m) * C(n-1, m-1)]? Wait, not sure. But in our case, it's 2 * [C(7,2)*C(7,1)].

Alternatively, maybe there's a different way to think about it. Let me consider that each direction change corresponds to a "turn". To have exactly four turns, you must have five straight segments. Each straight segment is a consecutive sequence of rights or ups.

To model this, suppose we have five segments. The first segment can be right or up. Let's say it's right. Then the segments alternate. So the segments are R, U, R, U, R. The total right steps: sum of the three R segments =8, each at least 1. Similarly, the U segments sum to 8, each at least 1. The number of ways to split the R steps is the number of positive integer solutions to r1 + r2 + r3 =8, which is C(8-1,3-1)=C(7,2). Similarly for the U segments: C(7,1). Multiply them together for the right-starting paths. Same for up-starting.

Therefore, the total is 2 * C(7,2)*C(7,1) = 2*21*7=294.

Therefore, I think that is the answer. But let me check for possible miscalculations.

Wait, 7 choose 2 is 21, 7 choose 1 is 7. 21*7 is indeed 147, times 2 is 294.

Alternatively, maybe the problem counts the starting direction as a direction change? Wait, no. The first move doesn't involve a direction change. The first direction change occurs after the first segment. So the count should be okay.

Alternatively, maybe the Asymptote examples have different structures. Let me check the first example path: (0,0) to (2,0) to (2,3) to (5,3) to (5,8) to (8,8). So the segments are:

Right 2, Up 3, Right 3, Up 5, Right 3. Wait, hold on, from (5,3) to (5,8) is up 5, then to (8,8) is right 3. Wait, so the last segment is right 3. Wait, but starting with right, the sequence should be R, U, R, U, R. So the last segment is right, which is correct. The lengths are 2,3,3,5,3. Wait, but the total right segments: 2+3+3=8? 2+3 is 5, plus 3 is 8. Yes. The up segments: 3+5=8. Correct. So that's a valid path.

Similarly the second example: (0,0) to (0,3) [up 3], then right 3 to (3,3), up 2 to (3,5), right 5 to (8,5), up 3 to (8,8). Wait, let's see:

Segments: Up 3, Right 3, Up 2, Right 5, Up 3. So starting with up, the sequence is U, R, U, R, U. The up segments: 3,2,3. Total 8. The right segments:3,5. Total 8. So that works.

Therefore, the examples are consistent with the formula.

Therefore, I think the answer is 294. But let me see if there is another way to get the same result.

Alternatively, perhaps using recursion or dynamic programming, but that might be more complicated. Alternatively, think of the problem as choosing points where the direction changes occur.

Suppose we model the path as a sequence of directions. Since we need exactly four direction changes, the direction sequence will alternate between right and up, starting with either. Let's consider starting with right first.

The direction sequence is R, U, R, U, R. So the direction changes occur after the first R segment, then after the first U segment, etc. The key is that between each pair of consecutive direction changes, the path moves in a straight line.

To model this, we can think of the path as being divided into five segments. The first segment is right, then up, then right, etc. The lengths of these segments must sum to 8 for rights and 8 for ups.

The number of ways to choose the lengths of the right segments is the number of ways to partition 8 into three parts (each at least 1), which is C(7,2). Similarly, partition 8 into two parts for the up segments, which is C(7,1). Multiplying gives the number for starting with right. Same for starting with up.

Therefore, this reinforces the previous result.

Alternatively, perhaps the problem is a standard combinatorial problem. I recall that the number of paths with exactly k turns is given by 2 * C(n-1, k/2)^2 when k is even? Wait, let me see.

Wait, in the case of an n x n grid, the number of paths with exactly 2m turns (so 2m+1 segments) is 2 * C(n-1, m)^2. Wait, let me test with our previous examples.

Take n=2, k=2 turns (so m=1). Then the formula would give 2*C(1,1)^2=2*1*1=2, which matches our earlier result.

For n=3, k=3 turns (so m=1.5?), no, wait k must be even? Wait, in the formula I just thought, maybe k is the number of turns, which must be even? Wait, in our problem, four direction changes is even. For the 3x3 grid with three direction changes (which is odd), the formula wouldn't apply. Hmm, perhaps my recollection is off.

Alternatively, perhaps the formula is for the number of turns, where a turn is a change from right to up or vice versa, so the number of turns is equal to the number of direction changes. Then, for an n x n grid, the number of paths with exactly k turns is 2 * C(n-1, floor(k/2)) * C(n-1, ceil(k/2)) or something.

Wait, in our case, n=8, k=4. Then floor(4/2)=2, ceil(4/2)=2. So it would be 2*C(7,2)*C(7,2). Wait, but that would be 2*21*21=882, which is different from our previous answer. Hmm, so that contradicts.

Alternatively, maybe my memory is incorrect. Let me think again.

Alternatively, perhaps the formula is as follows:

The number of paths from (0,0) to (a,b) with exactly k direction changes is equal to ... Hmm, perhaps I need to derive it.

Suppose we have a path from (0,0) to (m,n). The number of direction changes is equal to 2*t +1 or something? Wait, perhaps it's better to think in terms of the number of runs. A "run" is a consecutive sequence of the same direction. So the number of runs is equal to the number of segments, which is direction_changes +1.

Therefore, the number of paths with exactly k direction changes is equal to the number of paths with (k+1) runs.

The formula for the number of lattice paths with a given number of runs is known. Let me recall.

In general, the number of paths from (0,0) to (m,n) with exactly r runs in the x-direction (horizontal) is C(m-1, r-1)*C(n-1, r-1). Wait, maybe?

Wait, according to some combinatorial formulas, the number of lattice paths from (0,0) to (a,b) with exactly r "horizontal" runs (i.e., sequences of consecutive east steps) is C(a-1, r-1) * C(b-1, r-1). Because to have r horizontal runs, you need to partition the a steps into r parts, each at least 1, which can be done in C(a-1, r-1) ways, and similarly for the vertical runs. But the vertical runs would be r or r±1?

Wait, actually, if the path starts with a horizontal run, then the number of vertical runs is either equal to the number of horizontal runs or one less. Hmm, perhaps this is getting complicated.

Alternatively, the number of paths with exactly r horizontal runs and s vertical runs would be C(a-1, r-1)*C(b-1, s-1). But the total number of direction changes is (r + s -1). Because each run after the first requires a direction change.

Wait, if you have r horizontal runs and s vertical runs, the total number of runs is r + s, so the number of direction changes is (r + s -1).

Therefore, if we want exactly k direction changes, then r + s -1 =k, so r + s =k+1.

Additionally, if the path starts with a horizontal run, then the number of vertical runs can be r or r-1? Wait, actually, if starting with horizontal, then the sequence alternates H, V, H, V,... so if there are r horizontal runs, there are either r or r-1 vertical runs, depending on whether the number of runs is odd or even.

This is getting a bit too involved. Let me see if I can find a better approach.

Alternatively, let me think of the problem as follows. To have exactly four direction changes, the path must switch direction four times, leading to five segments. These five segments must alternate between horizontal and vertical.

Therefore, if the first segment is horizontal (right), then the segments are H, V, H, V, H. So three horizontal segments and two vertical segments. The total horizontal steps: sum of the three H segments =8, each at least 1. The vertical steps: sum of two V segments =8, each at least 1. The number of ways is the number of ways to partition 8 into three parts (for H) multiplied by the number of ways to partition 8 into two parts (for V).

Similarly, if the first segment is vertical, then we have three vertical segments and two horizontal segments.

The number of compositions (ordered partitions) of 8 into three parts is C(8-1,3-1)=C(7,2)=21. Similarly, compositions of 8 into two parts is C(8-1,2-1)=C(7,1)=7.

Therefore, the total number is 2*(21*7)=294.

Therefore, this seems consistent.

Alternatively, perhaps the formula for the number of compositions is the key here. A composition of an integer n into k parts is C(n-1,k-1). Since each segment must be at least 1, the number of ways to split the right steps into three segments is C(7,2), etc.

Therefore, I think the answer is 294.

Wait, but let me check once more with another perspective.

Suppose we model the path as a sequence of moves. Each direction change occurs when the direction switches. To have exactly four direction changes, the path must switch direction four times.

Let me think of the path as starting with a certain direction, then each switch alternates.

Let me consider the path starting with right. The directions go R, U, R, U, R. So the four direction changes occur between each pair of segments.

The positions where the direction changes occur can be thought of as points along the path where the direction switches.

Alternatively, the direction changes occur after certain steps.

But perhaps this is complicating.

Alternatively, think of the path as a sequence of steps. Each time you change direction, you must have moved at least one step in the previous direction.

Alternatively, the problem is analogous to arranging the steps with constraints on the number of runs.

In any case, given that multiple approaches lead to 294, and the small examples check out, I think the answer is 294.

However, just to be thorough, let me compute it numerically for a slightly bigger grid where I can count manually. Let's take a 3x3 grid, and compute the number of paths with four direction changes. Wait, four direction changes would mean five segments.

Starting with R: segments R, U, R, U, R. So three R segments and two U segments. The R total is 3, so compositions into three parts: C(3-1,3-1)=C(2,2)=1. Wait, 3 into three parts: only 1+1+1. The U segments must total 3 into two parts: C(3-1,2-1)=C(2,1)=2. So total starting with R:1*2=2. Similarly starting with U: compositions for U into three parts: C(2,2)=1, and R into two parts: C(2,1)=2. So total 2+2=4.

Wait, let's see. In a 3x3 grid, paths with four direction changes (five segments).

Starting with R:

The R segments must each be 1, since 1+1+1=3. The U segments can be split as (1,2) or (2,1).

So the paths would be:

1. R1, U1, R1, U2, R1.

2. R1, U2, R1, U1, R1.

Similarly starting with U:

U segments must be 1+1+1, and R segments split as (1,2) or (2,1).

So:

1. U1, R1, U1, R2, U1.

2. U1, R2, U1, R1, U1.

Thus total four paths. Which matches the calculation 2+2=4.

Therefore, the formula holds here as well.

Therefore, I feel confident that the answer is 294.

Wait, but hold on, the problem says "change direction exactly four times". Let me just confirm once again with the first example path. The first example has direction changes at (2,0) to up, then (2,3) to right, then (5,3) to up, then (5,8) to right. That's four direction changes. So that's exactly four.

Therefore, the formula counts those.

Therefore, I think the answer is 294.

But let me check once more the calculation:

C(7,2) is 21, C(7,1)=7. 21*7=147. Double that is 294.

Yes.

Therefore, I think the answer is 294.

**Final Answer**
\boxed{294}
</think>

To solve the problem of finding the number of paths from the lower left corner to the upper right corner of an \(8 \times 8\) grid that change direction exactly four times, we need to consider the structure of such paths.

A path that changes direction exactly four times will have five segments. These segments alternate between right (R) and up (U) moves. The path can start with either a right or an up move, leading to two cases:

1. **Starting with a right move (R):**
   - The direction sequence will be \(R, U, R, U, R\).
   - This results in three right segments and two up segments.
   - The total right moves must sum to 8, and the total up moves must also sum to 8.
   - The number of ways to partition 8 right moves into 3 segments (each at least 1) is given by \( \binom{7}{2} \).
   - The number of ways to partition 8 up moves into 2 segments (each at least 1) is given by \( \binom{7}{1} \).
   - Therefore, the number of paths starting with right is \( \binom{7}{2} \times \binom{7}{1} = 21 \times 7 = 147 \).

2. **Starting with an up move (U):**
   - The direction sequence will be \(U, R, U, R, U\).
   - This results in three up segments and two right segments.
   - The total up moves must sum to 8, and the total right moves must also sum to 8.
   - The number of ways to partition 8 up moves into 3 segments (each at least 1) is given by \( \binom{7}{2} \).
   - The number of ways to partition 8 right moves into 2 segments (each at least 1) is given by \( \binom{7}{1} \).
   - Therefore, the number of paths starting with up is \( \binom{7}{2} \times \binom{7}{1} = 21 \times 7 = 147 \).

Adding the two cases together, the total number of paths is \(147 + 147 = 294\).

Thus, the number of such paths that change direction exactly four times is \(\boxed{294}\).


Solution:
Let $z=a+bi$ such that $a^2+b^2=4^2=16$. The expression becomes:
\[(75+117i)(a+bi)+\dfrac{96+144i}{a+bi}.\]
Call this complex number $w$. We simplify this expression.
\begin{align*}
w&=(75+117i)(a+bi)+\dfrac{96+144i}{a+bi} \\
&=(75a-117b)+(117a+75b)i+48\left(\dfrac{2+3i}{a+bi}\right) \\
&=(75a-117b)+(116a+75b)i+48\left(\dfrac{(2+3i)(a-bi)}{(a+bi)(a-bi)}\right) \\
&=(75a-117b)+(116a+75b)i+48\left(\dfrac{2a+3b+(3a-2b)i}{a^2+b^2}\right) \\
&=(75a-117b)+(116a+75b)i+48\left(\dfrac{2a+3b+(3a-2b)i}{16}\right) \\
&=(75a-117b)+(116a+75b)i+3\left(2a+3b+(3a-2b)i\right) \\
&=(75a-117b)+(116a+75b)i+6a+9b+(9a-6b)i \\
&=(81a-108b)+(125a+69b)i. \\
\end{align*}
We want to maximize $\text{Re}(w)=81a-108b$. We can use elementary calculus for this, but to do so, we must put the expression in terms of one variable. Recall that $a^2+b^2=16$; thus, $b=\pm\sqrt{16-a^2}$. Notice that we have a $-108b$ in the expression; to maximize the expression, we want $b$ to be negative so that $-108b$ is positive and thus contributes more to the expression. We thus let $b=-\sqrt{16-a^2}$. Let $f(a)=81a-108b$. We now know that $f(a)=81a+108\sqrt{16-a^2}$, and can proceed with normal calculus.
\begin{align*}
f(a)&=81a+108\sqrt{16-a^2} \\
&=27\left(3a+4\sqrt{16-a^2}\right) \\
f'(a)&=27\left(3a+4\sqrt{16-a^2}\right)' \\
&=27\left(3+4\left(\sqrt{16-a^2}\right)'\right) \\
&=27\left(3+4\left(\dfrac{-2a}{2\sqrt{16-a^2}}\right)\right) \\
&=27\left(3-4\left(\dfrac a{\sqrt{16-a^2}}\right)\right) \\
&=27\left(3-\dfrac{4a}{\sqrt{16-a^2}}\right). \\
\end{align*}
We want $f'(a)$ to be $0$ to find the maximum.
\begin{align*}
0&=27\left(3-\dfrac{4a}{\sqrt{16-a^2}}\right) \\
&=3-\dfrac{4a}{\sqrt{16-a^2}} \\
3&=\dfrac{4a}{\sqrt{16-a^2}} \\
4a&=3\sqrt{16-a^2} \\
16a^2&=9\left(16-a^2\right) \\
16a^2&=144-9a^2 \\
25a^2&=144 \\
a^2&=\dfrac{144}{25} \\
a&=\dfrac{12}5 \\
&=2.4. \\
\end{align*}
We also find that $b=-\sqrt{16-2.4^2}=-\sqrt{16-5.76}=-\sqrt{10.24}=-3.2$.
Thus, the expression we wanted to maximize becomes $81\cdot2.4-108(-3.2)=81\cdot2.4+108\cdot3.2=\boxed{540}$.
~Technodoggo
Same steps as solution one until we get $\text{Re}(w)=81a-108b$. We also know $|z|=4$ or $a^2+b^2=16$. We want to find the line $81a-108b=k$ tangent to circle $a^2+b^2=16$.
Using $\frac{|ax+by+c|}{\sqrt{a^2+b^2}}=r$ we can substitute and get $\frac{|81(0)-108(0)-k|}{\sqrt{81^2+108^2}}=4$
\begin{align*} \frac{k}{\sqrt{18225}}&=4 \\\frac{k}{135}&=4 \\k&=\boxed{540} \end{align*}
~BH2019MV0
Follow Solution 1 to get $81a-108b$. We can let $a=4\cos\theta$ and $b=4\sin\theta$ as $|z|=4$, and thus we have $324\cos\theta-432\sin\theta$. Furthermore, we can ignore the negative sign in front of the second term as we are dealing with sine and cosine, so we finally wish to maximize $324\cos\theta+432\sin\theta$ for obviously positive $\cos\theta$ and $\sin\theta$.

Using the previous fact, we can use the [Cauchy-Schwarz Inequality](https://artofproblemsolving.com/wiki/index.php/Cauchy-Schwarz_Inequality) to calculate the maximum. By the inequality, we have:
$(324^2+432^2)(\cos^2\theta+\sin^2\theta)\ge(324\cos\theta+432\sin\theta)^2$
$540^2\cdot1\ge(324\cos\theta+432\sin\theta)^2$
$\boxed{540}\ge324\cos\theta+432\sin\theta$
~eevee9406
Similar to the solutions above, we find that $Re((75+117i)z+\frac{96+144i}{z})=81a-108b=27(3a-4b)$, where $z=a+bi$. To maximize this expression, we must maximize $3a-4b$. Let this value be $x$. Solving for $a$ yields $a=\frac{x+4b}{3}$. From the given information we also know that $a^2+b^2=16$. Substituting $a$ in terms of $x$ and $b$ gives us $\frac{x^2+8bx+16b^2}{9}+b^2=16$. Combining fractions, multiplying, and rearranging, gives $25b^2+8xb+(x^2-144)=0$. This is useful because we want the maximum value of $x$ such that this quadratic has real roots which is easy to find using the discriminant. For the roots to be real, $(8x)^2-4(25)(x^2-144) \ge 0$. Now all that is left to do is to solve this inequality. Simplifying this expression, we get $-36x^2+14400 \ge 0$ which means $x^2 \le 400$ and $x \le 20$. Therefore the maximum value of $x$ is $20$ and $27 \cdot 20 = \boxed{540}$
~vsinghminhas
First, recognize the relationship between the reciprocal of a complex number $z$ with its conjugate $\overline{z}$, namely:
\[\frac{1}{z} \cdot \frac{\overline{z}}{\overline{z}} = \frac{\overline{z}}{|z|^2} = \frac{\overline{z}}{16}\]
Then, let $z = 4(\cos\theta + i\sin\theta)$ and $\overline{z} = 4(\cos\theta - i\sin\theta)$.
\begin{align*} Re \left ((75+117i)z+\frac{96+144i}{z} \right) &= Re\left ( (75+117i)z + (6+9i)\overline{z}    \right ) \\                                                &= 4 \cdot Re\left ( (75+117i)(\cos\theta + i\sin\theta) + (6+9i)(\cos\theta - i\sin\theta)    \right ) \\                                                &= 4 \cdot (75\cos\theta - 117\sin\theta + 6\cos\theta + 9\sin\theta) \\                                                &= 4 \cdot (81\cos\theta - 108\sin\theta) \\                                                &= 4\cdot 27 \cdot (3\cos\theta - 4\sin\theta) \end{align*}
Now, recognizing the 3 and 4 coefficients hinting at a 3-4-5 right triangle, we "complete the triangle" by rewriting our desired answer in terms of an angle of that triangle $\phi$ where $\cos\phi = \frac{3}{5}$ and $\sin\phi = \frac{4}{5}$
\begin{align*} 4\cdot 27 \cdot(3\cos\theta - 4\sin\theta) &= 4\cdot 27 \cdot 5 \cdot (\frac{3}{5}\cos\theta - \frac{4}{5}\sin\theta) \\                                                &= 540 \cdot (\cos\phi\cos\theta - \sin\phi\sin\theta) \\                                                &= 540 \cos(\theta + \phi) \end{align*}
Since the simple trig ratio is bounded above by 1, our answer is $\boxed{540}$
~ Cocoa @ [https://www.corgillogical.com/](https://artofproblemsolving.comhttps://www.corgillogical.com/)
(yes i am a corgi that does math)
Follow as solution 1 would to obtain $81a + 108\sqrt{16-a^2}$.
By the Cauchy-Schwarz Inequality, we have
\[(a^2 + (\sqrt{16-a^2})^2)(81^2 + 108^2) \geq (81a + 108\sqrt{16-a^2})^2,\]
so
\[4^2 \cdot 9^2 \cdot 15^2 \geq (81a + 108\sqrt{16-a^2})^2\]
and we obtain that $81a + 108\sqrt{16-a^2} \leq 4 \cdot 9 \cdot 15 = \boxed{540}$.

- [spectraldragon8](https://artofproblemsolving.comhttps://artofproblemsolving.com/wiki/index.php/User:Spectraldragon8)
Follow solution 2 to get that we want to find the line $81a-108b=k$ tangent to circle $a^2+b^2=16$. The line turns into $a=\frac{k}{81}+\frac{4b}{3}$
Connect the center of the circle to the tangency point and the y-intercept of the line. Let the tangency point be $A$, the y-intercept be $C$, and the center be $B$. Drop the perpendicular from $A$ to $BC$ and call it $D$. Let $AD=3x$, $DC=4x$. Then, $BD=\sqrt{AB^2-AD^2}=\sqrt{16-9x^2}$. By similar triangles, get that $\frac{BD}{AD}=\frac{AD}{DC}$, so $\frac{\sqrt{16-9x^2}}{3x}=\frac{3x}{4x}$. Solve this to get that $x=\frac{16}{15}$, so $BC=\frac{20}{3}$ and $\frac{k}{81}=\frac{20}{3}$, so $k=\boxed{540}$
~ryanbear
Because $|z|=4$, we can let $z=4e^{i\theta}$. Then, substituting $i=e^{\frac{i\pi}{2}}$, we get that the complex number is
\begin{align*}
w&=4e^{i\theta}(75+117e^{\frac{i\pi}{2}})+\dfrac{1}{4}e^{-i\theta}(96+144e^{\frac{i\pi}{2}})\\
&=300e^{i\theta}+468e^{i(\frac{\pi}{2}+\theta)}+24e^{-i\theta}+36e^{i(\frac{\pi}{2}-\theta)}\\
\end{align*}
We know that the $\text{Re}(e^{i\alpha})=\cos(\alpha)$ from Euler's formula, so applying this and then applying trig identities yields
\begin{align*}
\text{Re}(w)&=300\cos{(\theta)}+468\cos{(\dfrac{\pi}{2}+\theta)}+24\cos{(-\theta)}+36\cos{(\dfrac{\pi}{2}-\theta)}\\
&=300\cos{(\theta)}-468sin{(\theta)}+24\cos{(\theta)}+36\sin{(\theta)}\\
&=324\cos{(\theta)}-432\sin{(\theta)}\\
\implies \dfrac{1}{108}\text{Re}(w)&=3\cos{(\theta)}-4\sin{(\theta)}\\
\end{align*}
We can see that the right-hand side looks an awful lot like the sum of angles formula for cosine, but 3 and 4 don't satisfy the pythagorean identity. To make them do so, we can divide everything by $\sqrt{3^2+4^2}=5$ and set $\cos{(\alpha)}::=\frac{3}{5}$ and $\sin{(\alpha)}::=\frac{4}{5}$. Now we have that
\[\dfrac{1}{540}\text{Re}(w)=\cos{(\theta+\alpha)}\]
Obviously the maximum value of the right hand side is 1, so the maximum value of the real part is $\boxed{540}$.
~Mooshiros
Let $c$ denote value of the above expression such that $\mathsf{Re} (c)$ is maximized. We write $z=4e^{i\theta}$ and multiply the second term in the expression by $\overline{z} = 4e^{-i\theta},$ turning the expression into
\[4e^{i\theta}(75+117i) + \frac{(96 + 144i)\cdot 4e^{-i\theta}}{4e^{i\theta}\cdot 4e^{-i\theta}} = 300e^{i\theta} + 468ie^{i\theta} + (24+ 36i)e^{-i\theta}.\]
Now, we write $e^{i\theta} = \cos\theta + i\sin\theta$. Since $\cos$ is even and $\sin$ is odd,
\begin{align*} &300(\cos\theta + i\sin\theta) +468i + (24+36i)(\cos\theta -i\sin\theta) \\ \iff & \mathsf{Re}(c) = 324\cos\theta -468\sin\theta \end{align*}
We want to maximize this expression, so we take its derivative and set it equal to $0$ (and quickly check the second derivative for inflection points):
\begin{align*}     &\mathsf{Re}(c) = 108\left(3\cos\theta - 4\sin\theta\right)\\     \frac{d}{d\theta} &\mathsf{Re}(c) = -324\sin\theta -468\cos\theta = 0, \end{align*}
so $\tan\theta = -\dfrac{468}{324} = -\dfrac{4}{3},$ which is reminiscent of a $3-4-5$ right triangle in the fourth quadrant (side lengths of $3, -4, 5$). Since $\tan\theta = -\frac{4}{3},$ we quickly see that $\sin\theta = -\dfrac{4}{5}$ and $\cos\theta = \dfrac{3}{5}.$ Therefore,
\begin{align*}  \mathsf{Re}(c) &= 108\left(3\cos\theta - 4\sin\theta \right) =  108\left(\frac{9}{5} + \frac{16}{5} \right)  = 108\cdot 5 = \boxed{\textbf{(540)}} \end{align*}
-Benedict T (countmath1)

Answer: 540

<｜begin▁of▁sentence｜><｜User｜>Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$.<｜Assistant｜><think>

Okay, so I need to find the largest possible real part of the expression (75 + 117i)z + (96 + 144i)/z where z is a complex number with |z| = 4. Hmm, let's see. Since |z| is given as 4, that means z lies on a circle in the complex plane with radius 4.

First, maybe I can represent z in terms of its polar form. Let me recall that any complex number z with |z| = 4 can be written as z = 4e^{iθ}, where θ is the argument of z. That might make it easier to handle the expression because then 1/z would be (1/4)e^{-iθ}, right? Since 1/z = conjugate(z)/|z|², so since |z|=4, 1/z = conjugate(z)/16. But maybe using exponentials is simpler here.

Let me write z = 4e^{iθ}, so 1/z = (1/4)e^{-iθ}. Then substituting into the expression:

(75 + 117i)z + (96 + 144i)/z = (75 + 117i)(4e^{iθ}) + (96 + 144i)( (1/4)e^{-iθ} )

Let me compute each term separately.

First term: (75 + 117i)*4e^{iθ} = 4*(75 + 117i)*e^{iθ}

Second term: (96 + 144i)*(1/4)*e^{-iθ} = (24 + 36i)*e^{-iθ}

So the entire expression becomes:

4*(75 + 117i)*e^{iθ} + (24 + 36i)*e^{-iθ}

Hmm, okay. Now, to find the real part of this expression. The real part of a complex number times e^{iθ} can be handled using Euler's formula. Let me recall that if I have a complex number A = a + bi, then A*e^{iθ} = (a + bi)(cosθ + i sinθ) = a cosθ - b sinθ + i(a sinθ + b cosθ). So the real part is a cosθ - b sinθ. Similarly for the second term.

Alternatively, maybe it's better to write both terms in terms of cosθ and sinθ. Let me try that.

Let me denote the first term as Term1 = 4*(75 + 117i)*e^{iθ}

Let me compute 4*(75 + 117i) first. That would be 4*75 = 300 and 4*117i = 468i, so Term1 is (300 + 468i)*e^{iθ}

Similarly, the second term is (24 + 36i)*e^{-iθ}

So, expanding both terms using Euler's formula:

Term1: (300 + 468i)(cosθ + i sinθ)

= 300 cosθ + 300i cosθ + 468i cosθ + 468i^2 sinθ

= 300 cosθ - 468 sinθ + i(300 sinθ + 468 cosθ)

Similarly, Term2: (24 + 36i)(cosθ - i sinθ) because e^{-iθ} = cosθ - i sinθ

= 24 cosθ - 24i sinθ + 36i cosθ - 36i^2 sinθ

= 24 cosθ + 36 sinθ + i(-24 sinθ + 36 cosθ)

So adding Term1 and Term2 together:

Real parts:

(300 cosθ - 468 sinθ) + (24 cosθ + 36 sinθ)

= (300 + 24) cosθ + (-468 + 36) sinθ

= 324 cosθ - 432 sinθ

Imaginary parts:

(300 sinθ + 468 cosθ) + (-24 sinθ + 36 cosθ)

= (300 sinθ -24 sinθ) + (468 cosθ + 36 cosθ)

= 276 sinθ + 504 cosθ

But since the problem asks for the real part, we can focus on the real part expression: 324 cosθ - 432 sinθ. We need to maximize this expression over all θ.

So the problem reduces to maximizing the function f(θ) = 324 cosθ - 432 sinθ.

Hmm, okay. To maximize this, I can recall that an expression of the form A cosθ + B sinθ can be written as C cos(θ + φ), where C = sqrt(A² + B²). Wait, but here it's A cosθ + B sinθ, but in our case it's 324 cosθ - 432 sinθ, so that's like A = 324, B = -432.

Alternatively, the maximum value of A cosθ + B sinθ is sqrt(A² + B²). So in this case, the maximum of 324 cosθ - 432 sinθ would be sqrt(324² + (-432)^2). Let me check that.

Yes, the maximum value of A cosθ + B sinθ is indeed sqrt(A² + B²). So here, A = 324, B = -432, so the maximum is sqrt(324² + 432²). Let me compute that.

First compute 324 squared:

324^2: 300² = 90000, 24²=576, and cross term 2*300*24=14400, so (300+24)^2 = 324² = 90000 + 14400 + 576 = 90000+14400 is 104400, plus 576 is 104,976.

Similarly, 432^2: Let's see, 400² = 160,000, 32²=1,024, and cross term 2*400*32=25,600. So (400+32)^2 = 432² = 160,000 + 25,600 + 1,024 = 160,000 +25,600 is 185,600 +1,024 is 186,624.

Therefore, sqrt(104,976 + 186,624) = sqrt(291,600). Hmm, sqrt(291600). Let me see, 540 squared is 291,600 because 500²=250,000 and 40²=1,600, and cross term 2*500*40=40,000, so (500+40)^2 = 540² = 250,000 + 40,000 + 1,600 = 291,600. So sqrt(291,600) is 540.

Therefore, the maximum real part is 540? Wait, but hold on. Let me confirm again. The expression is 324 cosθ - 432 sinθ, so A = 324, B = -432. Then sqrt(A² + B²) is sqrt(324² + 432²) which is sqrt(104,976 + 186,624) = sqrt(291,600) = 540. So yes, the maximum value is 540.

Therefore, the largest possible real part is 540? Hmm, but let me make sure I didn't skip any steps where maybe I made a mistake. Let me go through again.

First, expressing z as 4e^{iθ}, so 1/z is (1/4)e^{-iθ}. Then substituting into the original expression:

(75 + 117i)z becomes (75 + 117i)*4e^{iθ} = 4*(75 + 117i)e^{iθ}

Similarly, (96 + 144i)/z becomes (96 + 144i)*(1/(4e^{iθ})) = (24 + 36i)e^{-iθ}

Then expanding both terms:

First term expansion: (300 + 468i)(cosθ + i sinθ). The real part is 300 cosθ - 468 sinθ.

Second term expansion: (24 + 36i)(cosθ - i sinθ). The real part is 24 cosθ + 36 sinθ.

Adding the real parts: 300 cosθ - 468 sinθ +24 cosθ +36 sinθ = (300+24)cosθ + (-468 +36) sinθ = 324 cosθ - 432 sinθ. That's correct.

Then the maximum of that expression is sqrt(324² + 432²). Wait, but hold on, the formula is for A cosθ + B sinθ, which has maximum sqrt(A² + B²). Here, A is 324 and B is -432, so yes, sqrt(324² + (-432)^2) is the same as sqrt(324² + 432²). So that gives 540.

Therefore, the maximum real part is 540. Hmm, but let me check if there's another approach that might confirm this. Alternatively, maybe using calculus. Let me see.

Suppose we let f(θ) = 324 cosθ - 432 sinθ. To find its maximum, take derivative and set to zero.

f’(θ) = -324 sinθ - 432 cosθ. Setting to zero:

-324 sinθ - 432 cosθ = 0

=> 324 sinθ + 432 cosθ = 0

Divide both sides by cosθ (assuming cosθ ≠0):

324 tanθ + 432 = 0 => tanθ = -432/324 = -4/3 (since 432 divided by 324 is 4/3). So tanθ = -4/3.

Therefore, θ is in the second or fourth quadrant. Let me compute sinθ and cosθ. Let me suppose tanθ = -4/3, so we can think of a right triangle with opposite side -4 and adjacent 3 (or vice versa depending on the quadrant). The hypotenuse would be 5 (since 3-4-5 triangle).

So if tanθ = -4/3, then possible angles are θ = arctan(-4/3). Let me consider θ in the second quadrant where sinθ is positive and cosθ negative, or fourth quadrant where sinθ is negative and cosθ positive.

Wait, let's see. Let me suppose θ is in the fourth quadrant. Then cosθ = 3/5, sinθ = -4/5. Alternatively, in the second quadrant, cosθ = -3/5, sinθ = 4/5.

Let me plug into f(θ):

First, if θ is in the fourth quadrant: cosθ = 3/5, sinθ = -4/5.

Then f(θ) = 324*(3/5) -432*(-4/5) = (972/5) + (1728/5) = (972 + 1728)/5 = 2700/5 = 540.

If θ is in the second quadrant: cosθ = -3/5, sinθ = 4/5.

Then f(θ) = 324*(-3/5) -432*(4/5) = (-972/5) - (1728/5) = (-2700)/5 = -540. So that's the minimum.

Therefore, the maximum is indeed 540. So that's consistent with the earlier method.

Therefore, the answer should be 540. Wait, but let me check if there's any mistake in the arithmetic. Let me verify 324*(3/5):

324 divided by 5 is 64.8, times 3 is 194.4. 432*(4/5) is 432/5=86.4, times 4 is 345.6. Then 194.4 + 345.6 = 540. Yes, that's correct.

Alternatively, 324*3 = 972, 432*4 = 1728, so 972 + 1728 = 2700, divided by 5 is 540. Correct.

Therefore, the maximum real part is 540. Hmm, so that seems solid. But let me think again if there's another way to approach the problem, maybe using vectors or geometry?

Alternatively, since z is on the circle of radius 4, and 1/z is on the circle of radius 1/4. The expression is linear in z and 1/z. Maybe we can think of it as a combination of vectors?

Alternatively, perhaps write the expression as (75 + 117i) z + (96 + 144i)/z. Let me denote A = 75 + 117i and B = 96 + 144i. Then the expression is A z + B / z. Since |z|=4, then 1/z = \overline{z}/|z|² = \overline{z}/16. So maybe substituting that in:

A z + B * \overline{z}/16. Hmm, but not sure if that helps. Alternatively, perhaps write z = 4e^{iθ}, so \overline{z} = 4e^{-iθ}, so 1/z = e^{-iθ}/4. Wait, that's the same as before. So perhaps the first approach is the simplest.

Alternatively, maybe consider that the expression is A z + B / z, and since |z|=4, then we can write this as A z + B \overline{z}/( |z|² ). Since |z|² is 16, so it's A z + B \overline{z}/16. Hmm, but not sure if that helps. Maybe express in terms of z and its conjugate?

Alternatively, perhaps consider that for a complex number w, the real part of w is (w + \overline{w}) / 2. Maybe that could be a way to approach it, but perhaps more complicated.

Alternatively, think of the expression as a function on the circle |z|=4, and use calculus with complex variables? Maybe Lagrange multipliers? Let me see.

Let me consider z as a complex variable with |z|=4. Let me write z = x + yi, where x and y are real numbers satisfying x² + y² = 16. Then the expression becomes:

(75 + 117i)(x + yi) + (96 + 144i)/(x + yi)

First compute each term:

First term: (75 + 117i)(x + yi) = 75x + 75yi + 117i x + 117i^2 y

= 75x - 117y + (75y + 117x)i

Second term: (96 + 144i)/(x + yi). To simplify this, multiply numerator and denominator by the conjugate of the denominator:

(96 + 144i)(x - yi) / (x² + y²)

Since x² + y² = 16, so denominator is 16.

So numerator: 96x - 96yi + 144i x - 144i^2 y

= 96x + 144y + (-96y + 144x)i

Therefore, the second term is [ (96x + 144y) + ( -96y + 144x )i ] / 16

So combining both terms:

First term's real part: 75x - 117y

Second term's real part: (96x + 144y)/16 = 6x + 9y

Total real part: (75x - 117y) + (6x + 9y) = (75x +6x) + (-117y +9y) = 81x - 108y

Similarly, the imaginary parts would be (75y + 117x) + [ (-96y + 144x)/16 ] but since we only care about real part, we can focus on 81x -108y.

Wait, hold on, that's different from the previous result. Wait, in the first approach using polar coordinates, I got 324 cosθ - 432 sinθ, but here in Cartesian coordinates, I get 81x -108y. Hmm, that seems inconsistent. Wait, perhaps I made a mistake here.

Wait, let me check the second term again. The second term's real part:

The numerator after multiplying out was 96x +144y, so divided by 16 gives (96/16)x + (144/16)y = 6x + 9y. That's correct. Then adding to the first term's real part (75x -117y):

75x +6x = 81x, and -117y +9y = -108y. So total real part is 81x -108y.

But in the polar coordinate approach, I had 324 cosθ - 432 sinθ. Let me see if these are consistent. Since in polar coordinates, x = 4 cosθ, y = 4 sinθ. So substituting into 81x -108y:

81*(4 cosθ) -108*(4 sinθ) = 324 cosθ - 432 sinθ. Yes, exactly! So that matches. So both methods agree. Therefore, the real part is indeed 324 cosθ -432 sinθ, which has a maximum of 540.

Therefore, the answer is 540. Hmm, so that seems consistent across two different methods.

Alternatively, maybe I can think of 81x -108y with the constraint x² + y² = 16. To maximize 81x -108y subject to x² + y² =16. That's a standard constrained optimization problem. Using Lagrange multipliers:

Let f(x,y) =81x -108y, constraint g(x,y)=x² + y² -16=0.

The gradient of f is (81, -108), gradient of g is (2x, 2y). Setting gradient f = λ gradient g:

81 = 2λ x

-108 = 2λ y

So from first equation, λ = 81/(2x), from second, λ = -108/(2y) = -54/y. Therefore,

81/(2x) = -54/y => 81y = -108x => 3y = -4x => y = -4x/3.

Substitute into the constraint x² + y² =16:

x² + (16x²/9) =16 => (9x² +16x²)/9 =16 =>25x²/9 =16 =>x²= (16*9)/25 =>x= ±(12/5).

Therefore, x=12/5 or x=-12/5. Then y= -4x/3:

If x=12/5, then y= -4*(12/5)/3 = -16/5.

If x= -12/5, then y= -4*(-12/5)/3 = 16/5.

Now compute f(x,y)=81x -108y at these points:

First point (12/5, -16/5):

81*(12/5) -108*(-16/5) = (972/5) + (1728/5) = (972 +1728)/5 =2700/5=540.

Second point (-12/5,16/5):

81*(-12/5) -108*(16/5)= (-972/5) - (1728/5)= (-2700)/5= -540.

Therefore, the maximum is 540, which matches previous results.

Therefore, I think this is solid. The maximum real part is 540.

Wait, but just to be thorough, let me check with θ where tanθ = -4/3. Let me compute θ such that tanθ = -4/3. Let's pick θ in the fourth quadrant where cosθ is positive and sinθ is negative.

So if tanθ = -4/3, then sinθ = -4/5 and cosθ = 3/5 (since tanθ = sinθ/cosθ = -4/3, and sin²θ + cos²θ =1, so (16/9 cos²θ) + cos²θ =1 => (25/9 cos²θ)=1 => cos²θ=9/25, so cosθ= 3/5 or -3/5. Since in fourth quadrant, cosθ is positive, so 3/5, sinθ is -4/5.

Therefore, substituting into z =4e^{iθ}, so x=4cosθ= 4*(3/5)=12/5, y=4sinθ=4*(-4/5)= -16/5, which matches the Lagrange multiplier result. So that's consistent.

Therefore, all approaches lead to 540. Therefore, I think the answer is 540.

Wait, but let me check once again the initial substitution. Let me pick θ where tanθ = -4/3, so plugging back into the original expression, compute the real part. Let me compute z =4*(3/5 -4i/5)= 12/5 -16i/5.

Compute (75 +117i)z:

First, (75 +117i)*(12/5 -16i/5). Let me compute this:

Multiply 75*(12/5) = 75*(12)/5 = 15*12 = 180

75*(-16i/5) = 75*(-16)/5 i = -240i

117i*(12/5) = 117*12/5 i = (1404/5)i

117i*(-16i/5) = 117*(-16)/5 i² = (-1872/5)(-1) = 1872/5

So adding all terms:

Real parts: 180 + 1872/5 = (900 + 1872)/5 = 2772/5

Imaginary parts: -240i + (1404/5)i = (-1200/5 +1404/5)i = 204/5 i

Then compute (96 +144i)/z. Since z =12/5 -16i/5, so 1/z = conjugate(z)/|z|². |z|² is 16, so 1/z = (12/5 +16i/5)/16 = (12 +16i)/(5*16) = (12 +16i)/80 = (3 +4i)/20.

Therefore, (96 +144i)/z = (96 +144i)*(3 +4i)/20.

Compute numerator:

96*3 +96*4i +144i*3 +144i*4i

= 288 + 384i + 432i + 576i²

= 288 + (384+432)i +576*(-1)

= 288 + 816i -576

= (288 -576) +816i = -288 +816i

Divide by 20: (-288/20) + (816/20)i = -14.4 +40.8i

Therefore, adding to the first part:

First part's real part was 2772/5 = 554.4, and the second part's real part is -14.4. So total real part is 554.4 -14.4 = 540.

Yes! That's exactly 540. So that's a concrete example where the real part is 540. Therefore, this confirms that the maximum is indeed achievable and equals 540.

Therefore, after multiple methods and checks, I can confidently say the answer is 540.

**Final Answer**
\boxed{540}
</think>

Given the complex number \( z \) with \( |z| = 4 \), we need to find the largest possible real part of the expression \( (75 + 117i)z + \frac{96 + 144i}{z} \).

First, we represent \( z \) in polar form as \( z = 4e^{i\theta} \). Then, \( \frac{1}{z} = \frac{1}{4}e^{-i\theta} \).

Substituting \( z \) and \( \frac{1}{z} \) into the expression, we get:
\[
(75 + 117i) \cdot 4e^{i\theta} + (96 + 144i) \cdot \frac{1}{4}e^{-i\theta}
\]
Simplifying each term:
\[
4(75 + 117i)e^{i\theta} = (300 + 468i)e^{i\theta}
\]
\[
\frac{1}{4}(96 + 144i)e^{-i\theta} = (24 + 36i)e^{-i\theta}
\]

Expanding both terms using Euler's formula:
- For \( (300 + 468i)e^{i\theta} \):
  \[
  (300 \cos \theta - 468 \sin \theta) + i(300 \sin \theta + 468 \cos \theta)
  \]
- For \( (24 + 36i)e^{-i\theta} \):
  \[
  (24 \cos \theta + 36 \sin \theta) + i(-24 \sin \theta + 36 \cos \theta)
  \]

Combining the real parts:
\[
(300 \cos \theta - 468 \sin \theta) + (24 \cos \theta + 36 \sin \theta) = 324 \cos \theta - 432 \sin \theta
\]

To maximize \( 324 \cos \theta - 432 \sin \theta \), we use the fact that the maximum value of \( A \cos \theta + B \sin \theta \) is \( \sqrt{A^2 + B^2} \). Here, \( A = 324 \) and \( B = -432 \):
\[
\sqrt{324^2 + (-432)^2} = \sqrt{324^2 + 432^2} = \sqrt{104976 + 186624} = \sqrt{291600} = 540
\]

Thus, the largest possible real part of the given expression is \(\boxed{540}\).