Problem 23 · 2016 Math Kangaroo
Stretch
Number Theory
cryptarithmdigit-sum
The symbols ◯, □ and ◇ stand for three different digits. If the digits of the number ◯□◯ are added, you get the two-digit number □◇. If the digits of the two-digit number □◇ are added, you get the single-digit number □. Which digit does ◯ stand for?
Show answer
Answer: E — 9
Show hints
Hint 1 of 3
Start from the last clue: adding the two digits of □◇ gives □ again.
Still stuck? Show hint 2 →
Hint 2 of 3
If □ + ◇ = □, the diamond ◇ must be 0.
Still stuck? Show hint 3 →
Hint 3 of 3
Now use that the digits of ◯□◯ add to the two-digit number □0.
Show solution
Approach: chase the digit clues from the smallest one up
- The last clue says □ + ◇ = □, and the only way adding ◇ leaves □ unchanged is ◇ = 0, so the two-digit number □◇ ends in 0.
- The first clue says the digits of ◯□◯ add to that number, so ◯ + □ + ◯ = (a multiple of 10), i.e. 2◯ + □ is 10, 20, ...
- Trying □0 = 20 gives 2◯ + 2 = 20, so ◯ = 9, and the digits are all different (9, 2, 0).
- Check: 929 has digit sum \(9+2+9 = 20\), and \(2+0 = 2\) — it works, so ◯ = 9, choice (E).
Mark:
· log in to save