Geometric enumeration
Vanishing Squares
Investigate how many squares survive when dots disappear from a grid. A square is counted by its four corner dots, so a tilted square counts too. Your team records corner labels to avoid counting a square twice. Only pencil marks are removed; no cutting or physical counters are required.
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.
- Draw nine equally spaced dots in a 3×3 grid, labelled A B C across the top, D E F across the middle, and G H I along the bottom.
- Count a square whenever four remaining dots form its corners. Imaginary sides are allowed, a side may pass through another dot, and dots inside a square do not matter.
- Count each set of four corners once. Include tilted squares, but exclude non-square rectangles and diamond shapes whose corners do not have right angles.
Warm-up: count the full grid
With all nine dots present, list every square by its four corner letters.
| A | B | C |
| D | E | F |
| G | H | I |
Hint 1
Look for four small squares first.
Hint 2
The tilted square uses B, D, F and H.
Check the answer and reasoning
6 squares: ABDE, BCEF, DEGH, EFHI, ACGI, BDFH.
There are four small upright squares, one large outer square ACGI, and one tilted square BDFH. Writing sorted corner labels makes duplicate counting easier to spot.
Round 1: erase the centre
Cross out E. How many squares remain? List them; squares may surround a missing dot.
| A | B | C |
| D | × | F |
| G | H | I |
Hint 1
The centre is not a corner of the biggest square.
Hint 2
The tilted square also survives.
Check the answer and reasoning
2 squares: ACGI and BDFH.
Every small square needs E as a corner, so all four disappear. Neither the outer square nor the tilted square uses E as a corner.
Round 2: erase a corner instead
Start again with the full grid. Cross out A only. Which squares remain?
| × | B | C |
| D | E | F |
| G | H | I |
Hint 1
Compare with the six-square master list.
Hint 2
Cross out the two sets that contain A.
Check the answer and reasoning
4 squares: BCEF, DEGH, EFHI and BDFH.
Removing A destroys ABDE and ACGI. The other three small squares and the tilted square do not use A, so they remain.
Final: two opposite corners vanish
Restore the grid, then cross out A and I. List all surviving squares and explain why your list is complete.
| × | B | C |
| D | E | F |
| G | H | × |
Hint 1
Remove every set containing A or I.
Hint 2
A and I both affect the same large square.
Check the answer and reasoning
3 squares: BCEF, DEGH and BDFH.
From the original list, A removes ABDE and ACGI; I removes EFHI and ACGI. Count ACGI only once when removing it. The three untouched sets are BCEF, DEGH and BDFH.
Offer support
Connect one proposed set of corners lightly and compare all four side lengths. Start with the four small upright squares, then inspect the outer corners and the middle dots on each edge.
Finish together
Finish by listing every surviving square in the final position. Another player checks the list against the original six squares, crossing out only those that use a removed dot.
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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