Simultaneous sum constraints
Fifteen Square Lab
Complete a three-by-three number square whose rows, columns and two long diagonals all total fifteen. Each challenge has fixed clues and one completion in that orientation. Checking only the rows is not enough: your team must test all eight lines.
Work together, or form two small teams and rotate the writer. Age and time are planning estimates; choose the reading and arithmetic support that suit your group.
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Print one player sheet per team and one host guide for the adult. Both are black and white. Keep the host guide separate to avoid revealing answers. No cutting, folding, special props or account required.
No printer? Copy the grids and prompts onto plain paper. The full activity is below; hints and answers stay closed until selected.
Set up and rules
Bring pencils, an eraser and one sheet of paper per team. Print at 100% / actual size. Read the rules together, try the warm-up, work through two rounds, and finish with the final challenge.
- Use each whole number from 1 to 9 exactly once. Dots are empty boxes; keep every printed number in place.
- Every row, every column and both corner-to-corner diagonals must total 15.
- Draw the grid in pencil or write on the player sheet. Record a sum that forces a missing number before guessing.
Warm-up: find three gaps
Complete the grid. A dot is an empty box.
| 8 | · | 6 |
| 3 | · | 7 |
| 4 | · | 2 |
Hint 1
Start with the rows.
Hint 2
Each row has exactly one missing number.
Check the answer and reasoning
8 1 6 / 3 5 7 / 4 9 2
Top middle is 15−8−6=1; centre is 15−3−7=5; bottom middle is 15−4−2=9. The columns and diagonals then also sum to 15.
Round 1: use the centre
Complete this square while keeping 6 in the top-right corner.
| · | · | 6 |
| · | 5 | 7 |
| 4 | · | 2 |
Hint 1
The top-left to bottom-right diagonal is almost complete.
Hint 2
15−5−2=8.
Check the answer and reasoning
8 1 6 / 3 5 7 / 4 9 2
Top-left is 8 from the diagonal through 5 and 2. Top-middle is 1; middle-left is 3; bottom-middle is 9. The remaining lines confirm the arrangement.
Round 2: a new orientation
Complete the rotated grid. Do not copy the earlier rows without checking the printed clues.
| 4 | · | 8 |
| · | 5 | 1 |
| 2 | 7 | · |
Hint 1
Use the middle row first.
Hint 2
15−5−1=9.
Check the answer and reasoning
4 3 8 / 9 5 1 / 2 7 6
The middle row needs 9, the top row needs 3, and the bottom row needs 6. The top-right diagonal is 8+5+2=15.
Final: verify all eight lines
Complete the final grid, then write the totals of all three rows, three columns and two diagonals.
| 2 | 7 | · |
| · | 5 | 1 |
| 4 | · | · |
Hint 1
Finish the top and middle rows first.
Hint 2
Use the middle column to find the bottom-middle number.
Check the answer and reasoning
2 7 6 / 9 5 1 / 4 3 8
The top-right is 6, middle-left is 9, bottom-middle is 3 and bottom-right is 8. All eight sums are 15; the nine cells contain 1–9 once each.
Offer support
Provide the unused numbers as a written list. Work on a line with two known numbers first. An adult can handle subtraction while a younger player checks for duplicated numbers.
Finish together
The final square is complete only after the group checks three rows, three columns and two diagonals, and confirms that 1–9 each appears once.
If you disagree about an answer, return to the stated rules and compare the written steps. Keep the activity cooperative; a hint is part of play, and nobody needs to be eliminated.
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