Using the arrangement of letters in the margin, compute the number of paths that spell the word MATHEMATICS if all paths must start at the top and move diagonally down through the letters.

Question

Using the arrangement of letters in the margin, compute the number of paths that spell the word MATHEMATICS if all paths must start at the top and move diagonally down through the letters.

M
A A
T T T
H H H H
E E E
M M M M
A A A
T T
I I I
C C
S

I only got up to E E E, I don't know what to do after E E E

Answer 1

To continue from E E E, let's analyze the possible paths:

1. Path 1: From the rightmost E, you can move diagonally down-right to the first M.
2. Path 2: From the rightmost E, you can move diagonally down-left to the second M.

Let's examine each path:

Path 1:
- From the first M, you can move diagonally down-right to the first A.
- From the first A, you can move diagonally down-right to the first T.
- From the first T, you can move diagonally down-right to the H.
- From the H, you can move diagonally down-right to the second E.
- From the second E, there is no possible diagonal move. Therefore, this path cannot be continued to spell "MATHEMATICS."

Path 2:
- From the second M, you can move diagonally down-left to the second A.
- From the second A, you can move diagonally down-left to the second T.
- From the second T, you can move diagonally down-left to the first I.
- From the first I, you can move diagonally down-left to the C.
- From the C, you can move diagonally down-left to the S.

Therefore, there is only 1 valid path that spells "MATHEMATICS."

Please let me know if you need further assistance.

Answer 2

To compute the number of paths that spell the word MATHEMATICS, we can approach the problem recursively. Since all paths must start at the top and move diagonally down through the letters, we can keep track of our current position and the desired letter we want to reach.

Start at the top-left corner letter, 'M'. From 'M', we have three possible options to move diagonally down: 'A', 'T', and 'H'. Let's consider each option separately:

1. Moving to 'A':
From 'A', we have two possible options: 'T' and 'M'.

2. Moving to 'T':
From 'T', we have two possible options: 'H' and 'E'.

3. Moving to 'H':
From 'H', we have two possible options: 'E' and 'M'.

Now, let's continue the process:

4. From 'A', we have two possible options: 'T' and 'M'.
a. If we move to 'T', go to step 2.
b. If we move to 'M', go to step 3.

5. From 'T', we have two possible options: 'H' and 'E'.
a. If we move to 'H', go to step 3.
b. If we move to 'E', continue to the next step.

6. From 'E', we have two possible options: 'M' and 'T'.

Since you mentioned getting up to 'E E E', we can stop here.

Now, let's sum up the choices:
- Step 1: 3 options (A, T, H)
- Step 2: 2 options (T, M from A)
- Step 3: 2 options (E, M from H)
- Step 4: 2 options (T, M from A)
- Step 5: 1 option (E from T)
- Step 6: 2 options (M, T from E)

To compute the total number of paths, we multiply all the options together:
3 * 2 * 2 * 2 * 1 * 2 = 96

So, there are 96 paths that spell the word MATHEMATICS when moving diagonally down through the given arrangement of letters.

Answer 3

E E E \

M M M M /
A A A /
T T \
I I I/
C C/
S