Designing a 24-Point Worksheet in 10 Minutes With a Solver

The hand-picking step is what makes 24-point worksheets a pain. Pull four cards at random and you might land on 1 1 1 1 (unsolvable — the maximum reachable value with +, −, ×, ÷, and parens is 4), or 1 2 3 4 (eight solutions, every kid done in five seconds), or 3 3 8 8 (one solution and it runs through 1/3). A class of 9-year-olds will be either bored or stuck, often on the same page.

I have been helping my cousin's fourth-grade teacher build weekly 24-point worksheets for two months. The first one I built by hand took 45 minutes and the kids hated it. The second one took 10 minutes once I started using the 24-point solver as a sorter, and the kids finished the page about evenly. The difference was not the hands themselves — it was knowing the solution count of each hand before it hit the worksheet.

Why Random Hands Make Bad Worksheets

There are exactly 1,820 distinct multisets of four cards drawn from 1–13 (this is C(16, 4) by the stars-and-bars formula for choosing 4 elements with repetition from a 13-element set — verifiable on any combinatorics reference). The solution counts across those 1,820 hands are wildly uneven. A few have 20+ solutions. A few hundred have exactly one. Roughly 5% have none.

A teacher who picks 20 hands at random gets, in expectation, a mix that wastes about half the worksheet:

4–5 hands so easy the fastest student finishes in seconds
1–2 hands with no solution at all (kids stare at the page and lose confidence)
2–3 hands that require a fractional intermediate — beyond most fourth-graders
The remaining 10 or so hands are actually appropriate

The fix is not to pick fewer random hands. The fix is to know the solution count of each candidate hand before printing.

The Workflow: 30 Hands in Three Tiers

Here is the exact procedure I ran last Sunday for this week's worksheet.

Step 1. Open the solver in one tab, a spreadsheet in the other. Write down 50 candidate hands. I pick spreads that look interesting — 2 4 6 8, 7 7 7 7, 3 3 7 7, doubles, triples, runs. The exact list does not matter; the solver does the filtering.

Step 2. Paste each hand into the solver. Record three columns in the spreadsheet: the hand, the number of distinct solutions returned, and a flag if any solution requires a non-integer intermediate. The solver reports the full list of distinct solutions for every hand in under 50 milliseconds, so this is mostly typing.

Step 3. Apply the tier rules:

Tier A (warm-up, integer-only, ≥ 4 solutions). Examples: 1 2 3 4, 2 4 6 8, 1 3 5 9.
Tier B (main body, integer-only, 1–3 solutions). This is where the worksheet lives.
Tier C (challenge, single solution, requires a fraction). Examples: 3 3 8 8, 4 4 7 7, 1 5 5 5.

Step 4. Pick 10 from A, 15 from B, 5 from C. Shuffle within tiers. Print.

I keep a running master spreadsheet of about 400 classified hands and re-shuffle from it each week, so steps 1–3 only happen when I add new hands.

A Real Worksheet Row, Start to Finish

Take the hand 4 4 7 7. Paste it into the solver. One result comes back:

(4 − 4 ÷ 7) × 7 = 24

Step through it: 4 ÷ 7 = 4/7. 4 − 4/7 = 24/7. 24/7 × 7 = 24. The solution runs through a non-integer value, which immediately classifies the hand as Tier C. Most fourth-graders will not find it. That is exactly the right hand for the bottom of the worksheet — the kind that quietly tells the early-finishers "you are not done yet."

Compare with 2 4 6 8. Paste, run, and the solver returns multiple solutions, including the gentle (6 − 2) × 4 + 8 = 24. Check: 4 × 4 + 8 = 16 + 8 = 24. Every intermediate is a small integer. Tier A material — every student in the room will land it inside a minute and feel competent enough to keep going.

The point is not that I memorise these expressions. I do not — I forget the answer to 4 4 7 7 between worksheets. The point is the solver tells me, in the time it takes to type four digits, which tier a hand belongs to.

Three Pitfalls the Solver Catches

I have made all three of these mistakes before I had the solver in my workflow:

Including an unsolvable hand by accident. I once put 1 1 1 1 on a worksheet thinking the answer was (1 + 1 + 1) × 1 = 3 — and then realised the target is 24, not 3, and the hand has no solution at all. Five minutes lost in class explaining the mistake. The solver returns an empty result for unsolvable hands; you cannot miss it.

A hand whose only solution actually requires a fraction the kids have not seen. 1 5 5 5 looks innocent. The solver returns exactly one expression: (5 − 1 ÷ 5) × 5 = 24. That 1/5 intermediate is beyond what fourth-grade arithmetic has covered. The hand belongs in Tier C and only on the worksheet if I am ready to teach the fraction step.

A hand with so many solutions it is trivial. 1 2 3 4 has eight distinct solutions. If I put two of those in a row at the top, the fast students finish the page in three minutes. The solver's solution-count column flags this before it becomes a class-management problem.

For verifying any expression I cannot evaluate in my head — usually the fractional Tier C ones — I keep the scientific calculator open in a third tab. Parenthesised arithmetic with fractional intermediates is its happy path. For the rare hand where I want to double-check my arithmetic against a worked-out canonical example, the math formula reference covers basic fraction identities I have not used since university.

What the Solver Still Will Not Do

The one thing the solver does not grade is style. A student who solves 2 4 6 8 via (6 − 2) × 4 + 8 and another who solves it via 8 × 2 + 4 + 6 − 2… wait, that uses 2 twice — bad example, but you get the idea. The first student found something elegant; the second brute-forced it. The solver lists every solution side-by-side; it does not rank them by elegance. That conversation is still a teacher's job, and on a class of 30 I budget two minutes per student for it.

That is the trade I want from a tool. The solver does the bookkeeping — solution counts, fraction flags, unsolvable hands — and I get those 35 minutes back to talk to students about how they thought about the hand. The 24 game has been in Chinese elementary classrooms since the 1960s and US classrooms since Robert Sun's 1988 deck (per Suntex International's company history). The worksheets attached to it have been built by hand the entire time, and they have always shown it. They do not have to anymore.

Made by Toolora · Updated 2026-05-27