AMC 10 · 2007 · #9

Grade 4 rate-ratio
Archetype Modular Cycle Frequency Count └ Sub-archetype Calendar Day Lookup
triangular-numbersmodular-arithmeticpattern-recognition pattern-recognition ↑ Prerequisites: modular-arithmetic
📏 Medium solution 💡 3 insights
📘 View easy version →
Problem
A code replaces each letter of a message by a letter further right in the alphabet, shifting more every time that same letter comes back: the first time a letter appears it moves 1 place to its right, the second time 1+2 places, the third time 1+2+3 places, and so on, where A counts as one place to the right of Z. For example, "banana" becomes "cbodqg". In the message "Lee's sis is a Mississippi miss, Chriss!", which letter replaces the last s?

Pick an answer.

(A)
g
(B)
h
(C)
o
(D)
s
(E)
t

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The rule builds a clean pattern: the n-th s is pushed 1+2+...+n places right, a triangular number. And because the alphabet loops every 26 letters, only the shift's remainder after dividing by 26 matters. So the plan is to count the s's, find the running shift for the last one from the triangular pattern, then wrap it around the 26-letter cycle.

1STEP 1

Only the s's matter

A shift counts only that same letter's earlier appearances, so strip the message down to its s's and ignore every other letter.

2STEP 2

Count the s's

Tally the s's word by word: 1 + 2 + 1 + 4 + 2 + 2 = 12, so the last s is the 12th appearance.

1 + 2 + 1 + 4 + 2 + 2 = 12
3STEP 3

Find the 12th s's shift

The 12th appearance moves 1+2+3+...+12 places, a triangular number: 12 times 13 divided by 2 gives a shift of 78 places right.

1 + 2 + 3 + … + 12 = (12 · 13)/2 = 78
4STEP 4

Wrap around the 26-letter alphabet

The alphabet loops every 26 letters and 78 = 3 × 26, so the remainder is 0: the last s stays put, giving (D).

78 = 3 × 26 → 78 mod 26 = 0
Answer
s
A shift of 0 means the letter stays put, and s is one of the offered choices, so the result fits. The check also passes the sanity test on the given example: the 3rd a in "banana" shifts 1+2+3=6 places, giving g, exactly as "cbodqg" shows, so the same triangular-number rule that lands the last s on 78 is being applied correctly.
💡Key takeaway

Each letter counts only its own appearances, and since the alphabet loops every 26 steps, a shift that is a multiple of 26 leaves the letter unchanged.

  • Only the s's matter
  • Count the s's
  • Find the 12th s's shift
  • Wrap around the 26-letter alphabet

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