Three-pile misere strategy
Last Mark Strategy
Turn a historical take-away game into a paper strategy session. Cross out marks instead of handling matches. Players investigate which move leaves the other side without a winning reply. Losing a practice round supplies evidence for the next one; nobody is eliminated from the session.
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.
Get the complete free materials
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 separate rows of marks for the stated piles. Two players or two teams alternate turns.
- On a turn, cross out one or more remaining marks from exactly one pile. You may empty that pile, but may not touch another pile on the same turn.
- The player who crosses out the very last mark loses. A pile size of zero means that pile is empty. For a puzzle, name a winning first move and explain how to reply afterward.
Warm-up: two single marks
Piles are (1, 1). You move first. What should you cross out?
Hint 1
Do not take from both piles.
Hint 2
Leave exactly one mark.
Check the answer and reasoning
Empty either one pile, leaving one mark for the other player.
The opponent has only one legal move: take the last mark and lose. The two choices are equally good.
Round 1: a useful losing position
Piles are (1, 2). You move first. Which move guarantees a win against best replies?
Hint 1
Try leaving only one mark in total.
Hint 2
Taking both marks from one pile is legal.
Check the answer and reasoning
Remove both marks from the second pile, leaving (1, 0).
Leaving one mark forces the opponent to take the last. Removing only one from the second pile leaves two single marks, a winning position for the opponent.
Round 2: two piles of different sizes
Piles are (2, 3). You move first. Find the winning move.
Hint 1
Try making the piles equal.
Hint 2
Equal piles of 2 behave differently from equal piles of 1.
Check the answer and reasoning
Remove one mark from the pile of 3, leaving (2, 2).
If the opponent leaves (1,2), empty the pile of 2 and leave one mark. If the opponent leaves (0,2), remove one of those two. Both replies leave the opponent the last mark.
Final: three piles
Piles are (1, 2, 4). You move first. Find the winning move. Then test every kind of reply with the host.
Hint 1
There is one winning opening move.
Hint 2
Change only the largest pile, and leave it with 3.
Check the answer and reasoning
Remove one mark from the pile of 4, leaving (1, 2, 3).
The six replies leave (0,2,3), (1,1,3), (1,0,3), (1,2,2), (1,2,1), or (1,2,0). Reply respectively to reach (0,2,2), (1,1,1), (1,0,0), (0,2,2), (1,1,1), or (1,0,0). Each leaves the opponent a losing position.
Offer support
With 3–6 people, form two teams and rotate the person holding the pencil. First practise with one mark in each of two piles. Keep the final-mark-loses rule written beside the board.
Finish together
Finish by playing the final position twice with the teams swapping who moves first. Explain the winning reply rule; winning against a mistaken move alone is not a proof.
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.
Choose another paper puzzle · Try a shorter group game · Find a full mystery adventure