Problem 23 · 2024 Math Kangaroo
Stretch
Logic & Word Problems
caseworkcareful-counting
The picture shows a honeycomb with 16 cells. Some cells (but not all) are filled with honey. The number in a cell tells how many of its neighbouring cells are filled with honey. How many cells of the honeycomb are filled with honey?
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Answer: C — 9
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Hint 1 of 3
Start at the easiest clues: a cell saying 0 means none of its neighbours have honey, so colour all of them empty.
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Hint 2 of 3
A cell whose number equals how many neighbours it has must have every neighbour filled, so fill all of them.
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Hint 3 of 3
Keep going back and forth — each thing you mark empty or full gives new clues — until every cell is decided, then count the filled ones.
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Approach: start from the strongest clues and fill in the rest step by step
- A number in a cell just counts its honey-filled neighbours, so a 0 makes all its neighbours empty, and a number as big as the cell's neighbour-count makes them all filled.
- Begin with those certain cells, then each cell you settle tells you more about the cells touching it, like a chain of dominoes.
- Spreading these forced choices around the comb leaves exactly one pattern that fits every number.
- Counting the filled cells in that pattern gives 9.
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