Problem 25 · 2023 Math Kangaroo
Stretch
Counting & Probability
Logic & Word Problems
careful-countingcasework
Elisabeth wants to write the numbers 1 to 9 in the fields of the diagram shown so that the product of the numbers in any two fields next to each other is no greater than 15. Two fields are called “next to each other” if they share a common edge. How many ways are there for Elisabeth to label the fields?
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Answer: C — 16
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Hint 1 of 2
The big numbers (7, 8, 9) are very restricted: their neighbours' products must stay ≤ 15.
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Hint 2 of 2
Place 9, 8, 7 first into spots with few neighbours or small neighbours, then count the freedom that remains.
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Approach: place the large numbers under the product constraint, then multiply free choices
- For adjacent products ≤ 15, the largest numbers (9, 8, 7) can only sit next to very small numbers (mostly 1 and 2).
- This pins those big numbers to specific low-degree fields and forces small neighbours around them.
- Counting the independent choices that remain gives 16 labelings.
- So the answer is 16 (C).
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