2026 AMC 8, MOEMS & MATHCOUNTS Solutions

The 2026 elementary math competition season is wrapping up. Below are 8 selected problems from this year’s AMC 8, MOEMS Division E (grades 4–6), and MATHCOUNTS chapter round — the three flagship K-6 contests — solved with the Socratic thinking trace we use across the site.

Each problem links back to the prerequisite skill so a student who got stuck has a clear next step. No premium subscription, no walls — just the solutions and the reasoning behind them.

Why a thinking trace and not a clean solution? A clean solution shows what worked. A thinking trace shows what a strong solver tried first, what they noticed, and how they recovered. That’s the part that transfers to the next problem.

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AMC 8 (Grade 6 band)

Problem 1 — Counting paths on a grid

A frog at corner A of a 4×3 grid wants to reach corner B (the opposite corner), only moving right or up. How many distinct paths are there?

Setup. Any path is 7 moves: 4 rights and 3 ups, in some order.

Socratic trace.

“How many ways can I arrange 4 R’s and 3 U’s in a row?” “That’s 7-choose-3 — pick which 3 of the 7 positions are the U’s.” $\binom{7}{3} = \frac{7!}{3!,4!} = 35$.

Answer. 35 paths.

Prerequisite skill. Combinations and the multiplication principle. If your child stalled here, walk through Grade 6 expressions and equations to make sure variable thinking is solid before tackling combinatorics.

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Problem 2 — Average of consecutive integers

The average of the integers from 1 to n equals 51. Find n.

Socratic trace.

“Average of 1 to n is the middle value: $(n+1)/2$.” “So $(n+1)/2 = 51$, meaning $n = 101$.”

Answer. n = 101.

Prerequisite skill. Mean as the balance point — see Grade 5 line plots for the foundation.

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MOEMS Division E (Grades 4–6)

Problem 3 — Fraction comparison without LCM

Which is larger: $\dfrac{17}{30}$ or $\dfrac{12}{23}$? (No common denominator allowed.)

Socratic trace.

”$\frac{17}{30}$ is just over $\frac{1}{2}$ — specifically $\frac{1}{2} + \frac{2}{30} = \frac{1}{2} + \frac{1}{15}$.” ”$\frac{12}{23}$ is just over $\frac{1}{2}$ — specifically $\frac{1}{2} + \frac{1}{46}$.” ”$\frac{1}{15}$ > $\frac{1}{46}$, so $\frac{17}{30}$ wins.”

Answer. $\boldsymbol{\dfrac{17}{30}}$ is larger.

Prerequisite skill. Benchmark fractions — see Grade 4 comparing fractions.

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Problem 4 — Area dissection

A rectangle has area 60 cm². It is cut into 4 congruent smaller rectangles, each by halving one side and then halving the other. What is the perimeter of one small rectangle if the original is 10 cm × 6 cm?

Socratic trace.

“Halving 10 → 5; halving 6 → 3. Each small rectangle is 5 × 3.” “Perimeter = 2(5 + 3) = 16.”

Answer. 16 cm.

Prerequisite skill. Area and perimeter relationships — see Grade 3 area and Grade 3 perimeter.

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Problem 5 — Place value puzzle

A 4-digit number reads the same forwards and backwards. The sum of its digits is 22. How many such numbers are there?

Socratic trace.

“A palindrome 4-digit number has form $\overline{abba}$, so $a+b+b+a = 22$, i.e. $a+b = 11$.” “Pairs $(a,b)$ with $a \in {1\ldots9}$, $b \in {0\ldots9}$, $a+b=11$:” “(2,9), (3,8), (4,7), (5,6), (6,5), (7,4), (8,3), (9,2) — that’s 8 pairs.”

Answer. 8 numbers.

Prerequisite skill. Multi-digit place value — see Grade 4 place-value reasoning.

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MATHCOUNTS Chapter Round (Grades 6–8 band — accessible Grade 5+)

Problem 6 — Unit-rate reasoning

If 3 painters paint 5 walls in 4 hours, how long would 6 painters take to paint 20 walls (working at the same rate)?

Socratic trace.

“3 painters · 4 hours = 12 painter-hours per 5 walls.” “So 1 wall = 12/5 painter-hours.” “20 walls = 20 · 12/5 = 48 painter-hours.” “6 painters → 48 / 6 = 8 hours.”

Answer. 8 hours.

Prerequisite skill. Unit-rate reasoning — see Grade 6 ratios and unit rate.

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Problem 7 — Probability with paths

Two coins are flipped. What is the probability that at least one comes up heads?

Socratic trace.

“Total outcomes: HH, HT, TH, TT — 4 outcomes.” “At least one head: HH, HT, TH — 3 outcomes.” “Probability = 3/4.”

Answer. $\boldsymbol{\dfrac{3}{4}}$.

Prerequisite skill. Sample-space enumeration. Pair this with Grade 5 line plots for early data fluency.

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Problem 8 — Recursion in disguise

A staircase has 5 steps. You may climb 1 or 2 steps at a time. How many distinct ways can you climb the staircase?

Socratic trace.

“Let $f(n)$ = number of ways to climb $n$ steps.” “Base cases: $f(1) = 1$, $f(2) = 2$.” “Recurrence: $f(n) = f(n-1) + f(n-2)$ — Fibonacci!” “$f(3) = 3$, $f(4) = 5$, $f(5) = 8$.”

Answer. 8 ways.

Prerequisite skill. Pattern recognition — see Grade 5 patterns and rules.

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What to study next

If your child got stuck on 2 or more problems above, that’s a signal — not a verdict. Work back to the prerequisite skill:

• Stuck on counting/combinatorics? Build pattern fluency with Grade 5 patterns before re-attempting AMC-style counting problems.
• Stuck on fractions? Grade 4 comparing fractions and Grade 5 unlike denominators cover the highest-leverage skills.
• Stuck on rates/ratios? Move to Grade 6 expressions — every MATHCOUNTS rate problem reduces to one-step linear reasoning.
• Preparing for Math Kangaroo-style reasoning? Use the Math Kangaroo AI practice plan to keep hints helpful without answer-copying.
• Want low-screen practice? The travel math games post has 12 mental-math drills mapped to grade level.

Competition math rewards slow noticing over fast computation. The thinking traces above are deliberately verbose — that is the point. We will publish the 2027 recap on the same template next April.