Problem 21 · 2024 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintsubstitution
Mary wants to write the numbers 1 to 8 in the corners of the cube. For each of the six faces, the sum of the four numbers at its corners should be the same. She has already entered the numbers 6, 7 and 8 (see picture). What number does Mary have to write in the corner with the question mark?
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Answer: C — 3
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Hint 1 of 3
Add the eight corner numbers 1 + 2 + ... + 8 first, and notice every corner is shared by exactly 3 of the 6 faces.
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Hint 2 of 3
Use that to find what each single face must add up to, since all six faces have the same total.
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Hint 3 of 3
Then pick the one face that already shows known numbers next to the question mark and make it reach that total.
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Approach: find the common face-total, then fill in one face
- The corner numbers 1 through 8 add up to 36.
- Each corner sits on 3 of the 6 faces, so adding all six face-totals counts every corner 3 times: 3 × 36 = 108, which split over 6 equal faces makes each face add to 18.
- On the face that holds the question mark together with already-placed numbers, the four corners must reach 18, and filling in the known numbers leaves the question-mark corner as 3.
- So Mary writes 3 in the question-mark corner.
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