Problem 10 · 2012 Math Kangaroo
Medium
Algebra & Patterns
substitution
Alice and Bob send each other secret messages. To put their messages into code they use the following system: First each letter is given a number in order: A = 1, B = 2, C = 3, … Z = 26. Then the letter number is doubled and 9 is added. Bob received a message which began 19 – 37 – 48 – 19 – … Which of the following messages had Alice sent to Bob?
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Answer: E — Alice has made a mistake
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Hint 1 of 2
Undo the rule: from a received number, subtract 9 then halve to get the letter number.
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Hint 2 of 2
Check whether every received number actually gives a whole letter number.
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Approach: invert the encoding and test for validity
- The rule sends a letter number n to 2n + 9, so to decode you compute (received − 9) / 2.
- 19 → 5 = E and 37 → 14 = N, but 48 → (48 − 9)/2 = 19.5, which is not a whole number.
- Since 48 cannot come from any letter, the message could not have been encoded correctly.
- The answer is E: Alice has made a mistake.
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