AMC 10 · 2005 · #11
Grade 6 arithmeticPick an answer.
The phrase 'middle is the average' hides a cleaner rule. Turn it into an equation first, discover that only the first and last digits are free, then split the count into two tidy cases by parity and add them.
Turn the average into a sum
Clearing the fraction turns the condition into a sum.
The average of two numbers is just their sum split in half, so doubling it gives the sum back.
6.SP.A.3Convert To AlgebraThe middle digit takes care of itself
The middle digit is then forced and always legal.
Half of a number is a whole number only when that number is even, so a + c must be even.
2.OA.C.3Identify SubproblemsEven sum means matching parity
It is whole exactly when the two outer digits share parity.
Odd plus odd and even plus even both give even; a mismatch gives an odd sum.
Two digits add to an even number only when they are both odd or both even.
▸ Why?
Odd plus odd and even plus even both land on even, while a mismatched pair always lands on odd.
▸ Why?
The middle digit is the average of the outer two, and an average is a whole number only when the total is even.
Case 1 — both digits odd
Both odd gives 25 numbers.
Every odd first digit can be paired with every odd last digit, so multiply the choices.
3.OA.A.1Make A Systematic ListCase 2 — both digits even
Both even gives only 20, since the lead cannot be zero.
Zero is fine at the end but banned at the front, so the first digit loses one option.
4.NBT.A.2Make A Systematic ListAdd the cases
The cases do not overlap, so the count is 45, choice (E).
Separate, non-overlapping cases just get added together for the grand total.
4.NBT.B.4Make A Systematic ListThe middle digit being the average just means the first and last digits add to an even number, so count the matching-parity pairs and add the cases.
- Turn the average into a sum
- The middle digit takes care of itself
- Even sum means matching parity
- Case 1 — both digits odd
- Case 2 — both digits even
- Add the cases