[chelseapendley]

Given the set of letters below, you are to spell the word PASCAL by starting at the center of the rhombus and moving through the letters. Each move from center must be to the right, left, up or down (in other words, you may not move diagonally). How many ways can you find to spell the word PASCAL?

L
L A L
L A C A L
L A C S C A L
L A C S A S C A L
L A C S A P A S C A L
L A C S A S C A L
L A C S C A L
L A C A L
L A L
L

[ebaines]

This is pretty simple - how many ways do you get? Start with the P, then see how you would move to get to an "A," and then an "S," etc.

Your instructions say to start with a P at the center of "the rhombus," but you show it as a triangle. Does this show the problem correctly?

[chelseapendley]

I wrote it in a rhombus but it did it like this. I know its three digits the answer is

[chelseapendley]

It is suppose to look like this.

L
L A L
L A C A L
L A C S C A L
L A C S A S C A L
L A C S A P A S C A L
L A C S A S C A L
L A C S C A L
L A C A L
L A L
L

[ebaines]

OK - this makes more sense now. The way you first drew it there were only 2 ways, which really seemed too easy.

First thing to note is the symmetry of the arrangement - there are 4 quadrants that are identical, so one really only needs to consider one of the 4 quadrants and then multiply that result by 4.

I find it easied to do this backwards, Start with the letter L and work backwards toward the middle P. Again, by symmetry there are only 3 types of L's - the ones in the corners, those 1 position away from a corner, and those that are 2 positions away from a corner. Starting with the corner L's - you can see quite easily that there is only one path back to the center P with the letters in the correct order L-A-C-S-A-P. There are 4 corners L's, so that means you have 4 possible paths from all the corner L's. Now consider an L that's one removed from a corner - how many ways can you trace a path back to the P? I found 5 ways. There are 8 L's that are 1 spot from a corner, so that means these L's contribute a total of 40 paths to the answer. I'll leave it to you to figure out how many ways you can trace from any of the L's that's two positions away from a corner back to the center P, but here's a hint - it's more than 5. And the final answer does indeed turn out to be 3 digits.

One last thing - there's a huge hint in that the name you are spelling here is "Pascal." If you are familiar with Pascal's triangle, think about the fact that this puzzle has 6 terms in it - do you know what the coefficients are for the line of Pascal's triangle that has 6 terms? Add those coefficients up and you've got the number of paths in one of the four quadrants of this puzzle. Good luck!

[chelseapendley]

Is the answer 116 or 252?

[ebaines]

Neither one of those is the answer I got, but the first is pretty close. Why do you have two answers?

[chelseapendley]

I don't understand this question at all. I know nothing about the PASCAL triangle. Can you explain so I can understand?

[ebaines]

See: http://ptri1.tripod.com/

Note row 5 of the triangle - the one that goes 1,5,10,10,5,1.

I already showed you that in your problem the number of paths to a corner is 1, and the nunber of paths to an L one removed from a corner is 5. What do you think the number of paths to an L that's 2 off the corner is?

[chelseapendley]

9? Is the answer 120?

[ebaines]

I get:

(1 + 5 + 10+ 10+ 5) *4 = 124

[chelseapendley]

I still don't get how you get 1 5 10 10 and 5...

[ebaines]

Let's start with the first 1 - did you read my earlier post about how many ways there are to trace a path from the center P to one of the four corner "L's"? Post back and let me know if you get that, then we'll progress to the next term, which is a 5.