Problem 21 · 2024 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
There are exactly 2 frogs in each row and in each column (see picture). At the same moment, two of the six frogs each hop to an empty neighbouring square. Afterwards there are again 2 frogs in each row and in each column. In how many ways can two frogs hop like this?
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Answer: D — 4
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Hint 1 of 3
There are only three empty squares, so each hopping frog must land on one of those empty squares.
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Hint 2 of 3
When a frog leaves a row (or column) and another frog must keep that row (or column) at two, the two moves have to balance each other.
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Hint 3 of 3
Carefully try every pair of frogs that can hop into empty neighbours, and keep only the pairs that still leave two frogs in every row and every column.
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Approach: try each pair of frogs hopping into empty neighbours and keep the ones that stay balanced
- On the 3-by-3 grid the six frogs leave exactly three empty squares; a hopping frog can only move onto an empty neighbour.
- If one frog hops out of a row, the row drops to one frog, so a second frog must hop back into that same row to keep it at two — and the same must hold for columns.
- Go through the pairs of frogs that can both hop into empty neighbours and check which pairs keep every row and every column at two frogs.
- Exactly four such pairs of hops work, so the answer is 4 (D).
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