Problem 21 · 2014 Math Kangaroo
Stretch
Logic & Word Problems
caseworksum-constraint
Daniela fills a 3×3 table with the digits 1 to 9, one digit per cell. She has already placed 1, 2, 3, and 4 as shown. Two numbers are “adjacent” if their cells share a side. When she finishes, she notices that the numbers adjacent to 5 add up to 9. What is the sum of the numbers adjacent to 6?
| 1 | 3 | |
| 2 | 4 |
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Answer: E — 29
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Hint 1 of 2
The corners are fixed (1, 3, 2, 4); the centre and the four edge-middle cells still hold 5, 6, 7, 8, 9.
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Hint 2 of 2
An edge-middle cell touches two corners and the centre. Use 'the numbers next to 5 add to 9' to pin down 5 and the centre, then see what is next to 6.
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Approach: place 5 from its clue, which fixes the centre, then add 6's neighbours
- Corners: 1 (top-left), 3 (top-right), 2 (bottom-left), 4 (bottom-right). The centre and the four edge-middle cells take 5, 6, 7, 8, 9.
- An edge-middle cell is next to its two corners and the centre. For 5's neighbours to add to 9, the two corners by 5 must be small: only the left edge (corners 1 and 2) works, since 1 + 2 + centre = 9 gives centre = 6.
- So 6 sits in the centre, and the centre is next to all four edge-middle cells, which hold 5, 7, 8 and 9.
- The numbers adjacent to 6 add to 5 + 7 + 8 + 9 = 29.
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