Problem 20 · 2018 Math Kangaroo
Stretch
Logic & Word Problems
cryptarithmcasework
The symbols (an eye, a sun, an atom, a comb, and a fish) each stand for one of the digits 1, 2, 3, 4 or 5, all different. It is known that: atom + atom = fish, sun + sun = atom, and sun + fish = comb. Which symbol stands for the digit 3?
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Answer: A
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Hint 1 of 3
Each picture-symbol is a different digit from 1 to 5; start with the equation where the same symbol is added to itself.
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Hint 2 of 3
If a symbol doubled gives another symbol, that doubled symbol must be an even number, which narrows things down fast.
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Hint 3 of 3
Once you know four of the symbols, the leftover digit must belong to the fifth symbol.
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Approach: use the doubling equations to chain the symbols up to the digit 5, then the leftover digit names the eye
- From sun + sun = atom, the atom is double the sun; from atom + atom = fish, the fish is double the atom — so sun, atom and fish are 1, 2 and 4 (sun = 1, atom = 2, fish = 4), since doubling has to stay inside 1–5.
- From sun + fish = comb, the comb is 1 + 4 = 5, so the comb is 5.
- The digits 1, 2, 4 and 5 are now used up, so the only one left, 3, must be the remaining symbol, the eye — answer A.
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