Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #8
Grade 3 countingPick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only 3 × 3 = 9 ways to pick one chip from each bag, so Tool #11 (Make a Table) is the cleanest way to lay every sum out in a 3 × 3 addition grid — nothing can be missed. Tool #1 (Find a Pattern) then explains the structure of the answer: odd + even is always odd, and the sums are evenly spaced from the smallest (1+2=3) to the largest (5+6=11), so the distinct sums must be 3, 5, 7, 9, 11.
Build the addition table
Build a 3 × 3 addition table: Bag A values down the side, Bag B across the top, each cell the sum a + b.
An addition table is the Grade 3 way to organize "each from this set combined with each from that set" — every possible pair gets exactly one cell.
3.OA.A.1Work BackwardsRead off the nine sums
The table's nine sums are 3, 5, 7, 5, 7, 9, 7, 9, 11; collapsing the repeats leaves 3, 5, 7, 9, 11.
Spotting that sums repeat in a regular pattern — and that the set of different sums forms a Grade 3 arithmetic pattern — is the Tool #1 move.
The nine sums in the table are not nine different numbers; once the repeats are merged, exactly five distinct values remain, {3, 5, 7, 9, 11}.
▸ Why?
Several different chip pairs land on the very same total, so those cells are one value, not many — for instance 1+4 and 3+2 both make 5, and 1+6, 3+4, 5+2 all make 7.
▸ Why?
Stepping up to a Bag-A chip that is 2 larger while stepping down to a Bag-B chip that is 2 smaller just moves 2 from one addend to the other and leaves the total unchanged, since 3+4=(1+2)+4=1+(2+4)=1+6 — only the grouping of which parts are added first has changed.
▸ Why?
After the repeats are gone, the totals that survive are 3, 5, 7, 9, 11, and matching them one for one against 1, 2, 3, 4, 5 shows the collection holds five members.
Count the distinct sums
The set 3, 5, 7, 9, 11 has 5 members, so there are 5 different possible sums.
Counting the distinct entries that appear in the addition table is the same equal-groups counting move from Grade 3 multiplication.
3.OA.A.1Work BackwardsThis AMC 8 problem only needs Grade 3 skills — make an addition table, spot the pattern, and count — that you already know!
- Build the addition table
- Read off the nine sums
- Count the distinct sums
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