Case 10 · algebraic deduction · Medium · About 8–12 minutes
A toy workshop has two electronic scales and three sealed sample blocks. One scale is accurate. The other adds the same small amount to every displayed reading.
The technician records four measurements before forgetting which scale was faulty. Recover the block masses and identify the error.
The blocks are A, B, and C. Each has a positive whole-number mass from 1 to 20 grams.
The scales are G and H. Exactly one is accurate. The faulty scale always adds a fixed positive whole-number error of 1, 2, 3, 4, or 5 grams. It never subtracts. The error is added once per reading, regardless of how many blocks are weighed.
| Scale | Blocks weighed together | Displayed grams |
|---|---|---|
| G | A + B | 19 |
| G | B + C | 25 |
| H | A + C | 18 |
| H | A + B + C | 29 |
There are no containers, drifting readings, or other errors. The same blocks are used throughout.
What are the three block masses, which scale is faulty, and how much does it add?
Subtracting two readings from the same scale can be useful even before you know whether that scale is accurate. Its fixed error appears in both readings and cancels.
Hint 1 - reveal when you want a nudge
Subtract H’s A+C reading from H’s A+B+C reading. The fixed error cancels, leaving B’s actual mass.
Hint 2 - reveal when you want a nudge
B weighs 11 grams. Test H as the accurate scale first; then check whether making G accurate would force a forbidden negative error.
Reveal the solution and explanation
A weighs 6 g, B weighs 11 g, and C weighs 12 g. G is faulty and adds 2 g.
The H readings differ by 29 − 18 = 11. Adding B was the only change, so B weighs 11 grams whichever scale is faulty.
Suppose H is accurate. Then A + C = 18. If G adds e grams, its readings give A + e = 8 and C + e = 14. Adding those equations gives 18 + 2e = 22, hence e = 2. Thus A = 6 and C = 12.
For the other possibility, suppose G is accurate. Its readings then give A = 8 and C = 14. Their actual combined mass is 22, but H displayed 18. H would have to subtract four grams. The rules explicitly require a positive added error, so that possibility is impossible.
The surviving masses satisfy every reading and stay within the stated bounds. Both choices of faulty scale have been considered, proving the answer unique.
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