Off-by-one puzzles3 kinds
Name: Date:
Decide whether you want the things or the gaps. Show your working.
1A log is sawn into 14 equal pieces.How many cuts does that take?Why
2A log is sawn into 8 equal pieces.How many cuts does that take?Why
- Free
- Grade 2–4
- Challenge · two, three or four kinds
Things and the gaps between them worksheets
A log sawn into five pieces takes four cuts. Trees every five meters along a forty-meter path, one at each end, come to nine. Neither answer is the division — both are one away from it, in opposite directions.
- Answer key included
- Reopen this exact sheet
- Letter & A4 ready
Ready to printThings and the gaps between them
The answer key prints on a separate page.
What's on this sheet
There is almost no arithmetic here: one division at most, often none. The whole question is whether what is being counted is the things or the gaps between them, and which of those is one bigger. That is a decision rather than a calculation, and it is the earliest genuine decision this site can offer — a seven-year-old can meet it.
Why this sheet works
Four kinds of question share the page, and that is the design rather than variety for its own sake. Sawing a log gives one fewer cut than pieces. Trees with one at each end give one more tree than gap. Posts starting one gap in give exactly as many posts as gaps. A clock striking eight times has seven silences. A child who learns "add one" meets the log; a child who learns "subtract one" meets the path; no rule survives the sheet.