Competition · AMC preparation · step 4 of 4

AMC 8 · 2017 · #13

Grade 2 logicarithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
logical-deductionmental-arithmetic logical-deductionidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem
Peter, Emma, and Kyler play chess games against each other (no draws). Peter has 4 wins and 2 losses; Emma has 3 wins and 3 losses; Kyler has 3 losses. How many games did Kyler win?

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
4

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

If we draw each game as an arrow from the loser to the winner, every arrow contributes exactly one win and exactly one loss. The diagram makes the key invariant visible at a glance: the pile of wins and the pile of losses are the same size. Tool #1 (Draw a Diagram) turns an abstract counting argument into something you can literally see. Tool #6 (Guess and Check) lets us verify the answer against the multiple-choice list by plugging it back into the win-equals-loss condition.

1STEP 1

Pair every win with a loss

Draw each game as one arrow: it makes one win and one loss, so total wins always equal total losses.

Total Wins = Total Losses
2STEP 2

Add up all the losses

Add every loss the problem gives: 2 + 3 + 3 = 8 total losses.

2 + 3 + 3 = 8 total losses
3STEP 3

Add up the known wins

Add the wins we know: Peter's 4 + Emma's 3 = 7, with Kyler's W_K still missing.

4 + 3 + W_K = 7 + W_K total wins
4STEP 4

Set wins equal to losses

Use total wins = total losses: 7 + W_K = 8, so W_K = 8 - 7 = 1.

7 + W_K = 8 → W_K = 8 - 7 = 1
5STEP 5

Match to the choices

Kyler won 1 game — choice (B); testing the other choices in 7 + W_K = 8 confirms only this one works.

W_K = 1 → (B)
Answer
1
The answer W_K = 1 is between 0 and 4, well inside the answer-choice range. Sanity check the totals: wins = 4 + 3 + 1 = 8 and losses = 2 + 3 + 3 = 8 — they match, as they must when every game produces exactly one of each. Also, Kyler has more losses (3) than wins (1), which fits the story that the other two friends collectively won more games against him.
💡Key takeaway

This AMC 8 problem only needs Grade 2 addition and subtraction word-problem skills you already know!

  • Pair every win with a loss
  • Add up all the losses
  • Add up the known wins
  • Set wins equal to losses
  • Match to the choices

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