Term versus cumulative total
The Doubling Trail
Follow a trail whose numbered boxes double the marks at every step. The surprising part is the difference between the marks in the newest box and the marks accumulated across all boxes. Work with written numbers, so the session needs no pile of beans, counters or food.
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.
- Box 1 contains 1 mark. Each next box contains twice as many marks as the box before it.
- A box value means only that box. A running total includes every box from 1 up to the box named.
- Keep a two-column record: marks in the box, and marks so far. For large values, write numbers instead of drawing every mark.
Warm-up: four boxes
Write the value of boxes 1–4, then the total of all four boxes.
Hint 1
Double the newest box, not the whole running total.
Hint 2
Add all four box values for the final total.
Check the answer and reasoning
Box values: 1, 2, 4, 8. Total: 15.
The next box doubles the previous box: 1→2→4→8. Add 1+2+4+8 to get 15; the fourth box alone contains only 8.
Round 1: the sixth stop
What is in box 6, and how many marks are in boxes 1–6 altogether?
Hint 1
Continue the list by two boxes.
Hint 2
The total is one less than twice the newest box value.
Check the answer and reasoning
Box 6 has 32; the total is 63.
After box 4, the next values are 16 and 32. The six-box sum is 1+2+4+8+16+32=63.
Round 2: test a claim
A host says, “Box 8 contains 256 marks.” Give the correct box-8 value and the total for boxes 1–8.
Hint 1
Count the starting box before doubling.
Hint 2
Box 7 is 64.
Check the answer and reasoning
Box 8 has 128; the total is 255.
Box 1 starts at 1, so box 8 comes after seven doublings: 128. The next box, box 9, has 256. The sum through box 8 is 255.
Final: stay within 1000
A pretend budget allows 1000 marks in total. Starting at box 1, what is the greatest number of complete boxes you can include? Show why the next box cannot fit.
Hint 1
Do not compare 1000 only with the newest box.
Hint 2
Compare the running totals 511 and 1023.
Check the answer and reasoning
9 complete boxes: total 511. A tenth box raises the total to 1023, which exceeds 1000.
Box 9 holds 256, so the total through 9 is 255+256=511. Box 10 holds 512, taking the total to 1023. The unused 489 is less than the 512 needed for the next whole box.
Offer support
Use a calculator after box 6 if arithmetic becomes tiring. Cover the next row and predict it together. Emphasize that doubling a box value and adding to a running total are different steps.
Finish together
Finish by proving the final stopping point with two totals: the last affordable total and the very next total. A close guess without both checks is not enough.
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