Problem 22 · 2023 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
How many different ways are there to read the word BANANA in the following table if we can only cross to a field that shares an edge with the current field and we can use fields several times?
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Answer: E — 128
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Hint 1 of 2
Start on a corner B and spell B-A-N-A-N-A, stepping only to edge-adjacent cells (reuse allowed).
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Hint 2 of 2
Count routes by multiplying the choices at each letter; symmetry across the four corners helps.
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Approach: multiply the branching choices, letter by letter
- A path spells B-A-N-A-N-A, moving each step to an edge-adjacent cell, and cells may be reused.
- Start from a B and count how many neighbours carry the next needed letter at each move; the number of full paths is the product of those branch counts along the route.
- Because the grid is symmetric, every starting B contributes the same number of paths, so multiply one B's count by the number of B's.
- Carrying out this branch-multiplication over the whole grid totals 128 ways (choice E).
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