Reversing an operation chain
Number Machine Detectives
A number machine seems to reveal a secret starting number. Your family can work out how by recording its operations and reversing them. The puzzle uses arithmetic rather than mind reading, and every answer can be checked by running it through the same machine.
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.
- Choose a whole starting number from 1 to 20. Double it, add 6, halve the result, then multiply by 4.
- Write down every intermediate value. Do operations in the stated order; do not combine different players’ records.
- To work backward, reverse the order and undo each operation: divide by 4, double, subtract 6, then halve.
Warm-up: travel forward
Start with 5. Double, add 6, halve, then multiply by 4. Record all four results.
Hint 1
Follow just one arrow at a time.
Hint 2
The third result is half of 16.
Check the answer and reasoning
10 → 16 → 8 → 32
Doubling 5 gives 10; adding 6 gives 16; halving gives 8; multiplying by 4 gives 32.
Round 1: recover the start
A player reports a final output of 48. Which whole number did they start with?
Hint 1
Undo the last operation first.
Hint 2
Begin with 48÷4.
Check the answer and reasoning
9
48÷4=12; 12×2=24; 24−6=18; 18÷2=9. Forward check: 9→18→24→12→48.
Round 2: catch an impossible report
A player reports an output of 42, while claiming a whole starting number from 1 to 20. Can that be correct?
Hint 1
Try the reverse route even if a fraction appears.
Hint 2
42÷4=10.5; continue from there.
Check the answer and reasoning
No; reversing 42 gives 7.5, not a whole number.
The machine output is 4 times the starting number plus 12. Outputs from whole numbers are multiples of 4. 42 is not a multiple of 4; a calculation was recorded incorrectly.
Final: repair the record
Two outputs are 64 and 88. Recover both starts, then run each one forward to verify all four intermediate values.
Hint 1
Subtract 12 from each output, then divide by 4.
Hint 2
The two starts are 13 and 19; now prove them by going forward.
Check the answer and reasoning
64 comes from 13; 88 comes from 19.
For 13: 26→32→16→64. For 19: 38→44→22→88. The inverse rule gives the same starts: (output−12)÷4.
Offer support
Use a calculator for arithmetic if needed, while players decide which operation comes next. Label the four boxes so nobody has to remember the whole sequence.
Finish together
Finish when the group can explain the reverse route and use it on both final outputs. Ask why an output of 42 cannot come from a whole starting number in this machine.
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