Problem 19 · 2013 Math Kangaroo
Hard
Number Theory
sum-constraintcasework
Numbers are written in the 4 × 4 grid so that any two numbers in squares sharing an edge differ by 1. The number 3 is already given, and the number 9 is used somewhere in the grid. How many different numbers are used once the grid is completely filled in?
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Answer: D — 7
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Hint 1 of 3
Touching squares differ by exactly 1, so as you step across the grid the number only goes up or down by 1 each time.
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Hint 2 of 3
To get from a 3 to a 9 you must pass through every value in between.
Still stuck? Show hint 3 →
Hint 3 of 3
List the numbers you are forced to step through from 3 up to 9 and count them.
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Approach: track the range of values forced by the rules
- Moving from one square to a touching square changes the number by just 1, so to climb from 3 up to 9 you must step through 4, 5, 6, 7 and 8 along the way.
- That forces all of 3, 4, 5, 6, 7, 8, 9 to appear in the grid.
- So 7 different numbers are used, which is choice D.
Mark:
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