Problem 11 · 2023 Math Kangaroo
Medium
Number Theory
Logic & Word Problems
place-valuenumber-systems
Dorli writes down three consecutive natural numbers in increasing order. She replaces the digits with symbols and gets: □♦♦, ♡△△, ♡△□. What would be the next bigger number in this notation?
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Answer: E — ♡△♡
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Hint 1 of 3
Each symbol is one fixed digit, and the three codes are consecutive numbers, so the jump from the first to the second is just adding 1.
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Hint 2 of 3
Notice the hundreds symbol changes between the first and second number — that signals a roll-over like 199 → 200.
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Hint 3 of 3
Once you know what each symbol stands for, write the third number, add 1 more, and re-encode it.
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Approach: decode the symbols using the roll-over, then add one more
- From the second to the third number only the last digit changes (♡△△ → ♡△□), so adding 1 turns △ into □, meaning □ = △ + 1 with no carry.
- From the first to the second number the hundreds digit changes (□♦♦ → ♡△△), which only happens on a roll-over like 199 → 200: so ♦ = 9, △ = 0, □ = 1 and ♡ = 2.
- The three numbers are 199, 200, 201, and the next bigger one is 202.
- Re-encoding 202 with 2 = ♡ and 0 = △ gives ♡△♡, option E.
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