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2017 Math Kangaroo

Problem 15

Problem 15 · 2017 Math Kangaroo Stretch
Logic & Word Problems cryptarithmwork-backward

Each one of the four keys locks exactly one padlock. Every letter on a padlock stands for exactly one digit, and the same letters mean the same digits. Which letters must be written on the fourth padlock?

Figure for Math Kangaroo 2017 Problem 15
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Answer: D — GAG
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Hint 1 of 2
Match each key's number to the padlock it opens, lining up the digits with the letters in the same spots.
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Hint 2 of 2
Once you know which digit each letter is, the leftover key tells you the fourth padlock.
Show solution
Approach: decode letters to digits, then read the leftover key
  1. The lock ADA has its first and last letters the same; the only key like that is 141, so A=1 and D=4.
  2. Now DAG starts 4, 1, so its key is 417, giving G=7; then DGA is 4, 7, 1, which is the key 471.
  3. Those three keys are used, so the fourth padlock is opened by the leftover key 717.
  4. Reading 717 back as letters gives 7=G, 1=A, 7=G, so the fourth padlock is GAG (option D).
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