Problem 10 · 2025 Math Kangaroo
Medium
Counting & Probability
caseworkcareful-counting
Vera has built a tower of cubes. She wants to replace the two cubes with question marks with cubes with numbers. Vera wants the number on each cube to be at least 2 higher than the number on the cube below it. How many ways can she replace the two cubes?
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Answer: D — 6
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Hint 1 of 2
The two missing cubes sit above 6 and below 14, each at least 2 more than the one below.
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Hint 2 of 2
List the lower missing number first, then count valid choices for the upper one.
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Approach: count valid number pairs by casework
- Reading up, the tower is 1, 4, 6, then two unknowns, then 14, each cube at least 2 more than the one below.
- The lower unknown must be at least 8 (it is at least 2 more than 6).
- The upper unknown is at least 2 more than it and at most 12 (since 14 is at least 2 more).
- Counting all such pairs gives 6 ways.
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