Problem 21 · 2018 Math Kangaroo
Stretch
Number Theory
place-value
Three different digits A, B, and C are chosen. Then the biggest possible six-digit number is built in which the digit A appears 3 times, the digit B 2 times, and the digit C 1 time. Which arrangement is definitely not possible for this number?
Show answer
Answer: D — AAABCB
Show hints
Hint 1 of 2
The biggest number puts its six digits in order from largest on the left to smallest on the right, so reading left to right the digits never go back up.
Still stuck? Show hint 2 →
Hint 2 of 2
Watch the two copies of B: if any other letter sits between them, that letter would have to be no bigger than B and no smaller than B at the same time, forcing it to equal B - which is not allowed.
Show solution
Approach: the biggest number lists its digits from largest to smallest, so equal digits must be together
- To make the number as big as possible you write its six digits from largest to smallest, so going left to right the digits never increase.
- Then the two B's must sit side by side: any letter caught between them would have to be no bigger than B (to its left) yet no smaller than B (to its right), so it would equal B - impossible, since the digits are all different.
- In AAABCB the C sits between the two B's, which can never happen.
- Every other option keeps the two B's together, so the impossible one is AAABCB.
Mark:
· log in to save