Problem 15 · 1992 AJHSME
Hard
Number Theory
periodic-sequencemod-arithmetic
What is the 1992nd letter in the sequence ABCDEDCBAABCDEDCBAABCDEDCBA… ?
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Answer: C — C.
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Hint 1 of 3
The pattern just repeats the same chunk over and over. Find where one chunk ends and the next begins — how many letters is one full chunk?
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Hint 2 of 3
For a repeating pattern, only the LEFTOVER after removing whole chunks matters: chop off complete blocks, and the position you land on inside the last block is your answer.
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Hint 3 of 3
Divide 1992 by the block length and look at the leftover, not the quotient — the leftover tells you how far into a fresh block you are.
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Approach: strip away whole blocks; the leftover position names the letter
- The repeating chunk is ABCDEDCBA, which is 9 letters long. After that it starts over.
- Take away as many whole 9-letter blocks as fit in 1992: 9 × 221 = 1989, leaving 1992 − 1989 = 3 letters into the next block.
- The 3rd letter of ABCDEDCBA is C.
- Why this transfers: every "which item is in spot n" pattern problem works this way — the answer depends only on the leftover after removing complete cycles, never on how many cycles there were. A leftover of 0 would mean the very last letter of a block.
- Sanity check: spots 1, 2, 3 are A, B, C; since 1992 is 3 past a clean block boundary, it must match spot 3 = C.
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