Three Level Scales Riddle: How Many Circles Balance the Last?
This article was written with AI and may contain mistakes. Found one? Tell us.
The last scale needs 5 circles. The three level scales show that a square weighs the same as 2 circles and a triangle the same as 3 circles, so a triangle and a square together weigh 5 circles.
How do the three scales give the answer?
A level scale means both pans weigh the same, so each scale is a small equation. Read them one at a time:
- Scale one: a triangle balances a square and a circle.
- Scale two: two squares balance a triangle and a circle.
- Scale three: two triangles balance three squares.
Now swap the triangle on scale two for what scale one says it is worth. Two squares then balance a square and two circles. Take one square off each pan, and a single square balances two circles. Scale one then says a triangle is a square plus a circle, which makes three circles.
Scale three is your check. Two triangles are six circles, and three squares are also six circles, so it stays level. The last pan holds one triangle and one square: three circles plus two circles is five.
What are the usual wrong answers?
| Wrong answer | Why it feels right | Why it fails |
|---|---|---|
| 2 circles | Two shapes on the pan, one circle each | The shapes are heavier than a circle |
| 3 circles | The triangle's weight is found first | The square on the same pan is forgotten |
| 6 circles | Scale three adds up to six | That pan holds two triangles, not one triangle and a square |
Why do balance puzzles feel so fair?
A balance gives an exact comparison with only three possible results: the left pan is lighter, the right pan is lighter, or the pans stay level [1]. The simplest classic coin puzzle asks you to find one lighter coin among coins that look the same, using as few weighings as possible, and it first appeared in 1945 [1].
A 2020 paper on logic puzzles calls a puzzle good when it has exactly one solution [2]. This riddle passes that test: the three scales fix the square and the triangle completely, and no other number keeps the last scale level.
Ready for another picture riddle?
Each riddle below turns a picture into a few exact rules.
References
- Khovanova, T., & Knop, K. (2025, June 26). Coins that Change Their Weights. Recreational Mathematics Magazine, 12(20), 65-101. DOI 10.2478/rmm-2025-0005
- Alzboon, L., & Nagy, B. (2020, February 4). Truth-Teller–Liar Puzzles with Self-Reference. Mathematics, 8(2), 190. DOI 10.3390/math8020190