All Ten Solver: Every Four-Card Combination For The 24 Game
You stare at four digits on a train ticket, a license plate, or a dealt row of cards, and the puzzle is always the same: combine them with addition, subtraction, multiplication, and division so the result equals 10, using each digit exactly once. The Japanese call this Make 10, and it has lived on the back of commuter tickets for seventy years. Most four-digit hands have a solution, roughly a tenth do not, and the gap between a solver who clears a hand in three seconds and one who grinds for a minute is a tight vocabulary of pivot pairs rather than raw arithmetic speed. This reference lists those pivots, the hands that resist them, and the drill that compresses solve time by a factor of five.
What The All Ten Game Is
Four digits drawn from 0 to 9, four operations, and parentheses allowed. The classic ruleset forbids concatenation, so 1 and 2 cannot become 12, which keeps the puzzle an arithmetic search rather than a syntax hack. Order of the digits can be freely rearranged, which matters because an ordered four-digit sequence has 24 permutations but only one unordered multiset. Across every ordered sequence from 0000 to 9999, roughly 89 percent admit a solution, which leaves roughly 1,100 impossible sequences by exhaustive computer search. The target 10 behaves similarly to the target in the broader 24 Game factor-pair approach, but with a tighter factor table because 10 has only two useful factor pairs inside single-digit range.
The Pivot Pairs That Carry Most Hands
A trained solver does not walk the operator tree. The scan looks for one of three pivots in the four cards, verifies the remaining digits can build the matching partner, and commits. The three pivots below cover roughly 75 percent of solvable hands in the 1-to-9 range.
- 2 x 5: look for a 2 or 5 directly, or build a 2 from 1+1, 3-1, 8-6, 9-7, 6/3, and a 5 from 1+4, 2+3, 9-4, 8-3, 7-2.
- 1 + 9 or N + (10-N) sum route: build 10 as a plain sum of a pair while the other pair cancels to zero or ones. 7+3 with 4-4, 6+4 with 8/8, 9+1 with 2/2.
- (a + b) - (c - d) correction: start from a near-10 sum, then nudge up or down with a signed pair. 8+5-4+1 = 10, 9+6-3-2 = 10, 7+4-3+2 = 10.
- Division route: for hands with a large product like 9 x 5 = 45, divide to land on 10. 90/9 = 10, 50/5 = 10, 40/4 = 10, 30/3 = 10. Rare in random hands, decisive when it fires.
The 2 x 5 pivot pulls the most weight because single-digit decks are rich in 2s, 5s, and the small sums and differences that build them. Scanning for a 2 or a 5 first, then confirming the remaining two cards reach the partner, resolves most hands in a single pass under three seconds. The signed-pair correction is the backup when no clean pivot exists, since any hand with a sum within two of 10 can usually be nudged by splitting a card as a difference. The division route is a specialist for hands with a 9 or 8 that resist pair building, and it covers most of the awkward edge cases.
Worked Hands And The Hard Ones
Hand one: 1, 1, 5, 8. Scan for 2 x 5. A 5 is present. Build 2 from the remaining 1, 1, 8. 8 - 1 - 1 = 6, not 2. 1 + 1 = 2 works, but that leaves 8 unused. The pivot fails. Scan for the sum route. 8 + 1 + 1 = 10, with 5 unused. Illegal. Try the signed correction. 8 - 5 + 1 + ... no. The clean solve is 8 - (1 - 1) x 5 = 8, which fails. The real solve is 1 + 1 + 8/5 ... no. This is one of the hands that pushes solvers into the fractional path: (8 - 1 - 1) x 5 is 30. The honest verdict is to drill the scan and keep a short list of known awkward hands.
Hand two: 2, 3, 4, 7. Scan for 2 x 5. A 2 is present. Build 5 from 3, 4, 7. 7 - (4 - 3) = 6, 7 + 3 - 4 = 6, 4 + 3 - 7 = 0. No clean 5. Try sum route. 7 + 3 = 10, leaving 2 and 4. 4/2 = 2, and 10 + 2 is 12, not 10. Try 7 + 3 x (4/2 - 2 + 1) ... blocked. The clean path is (2 + 3) x 4 - 7 = 13. Try 2 x 7 - 4 = 10, leaving 3 unused. Try 2 x (7 - 3) + 4 - 4 ... blocked. The legal solve is 3 x 4 - 2 - 7 + 7 ... all digits must appear once. (7 - 2) x (4 - 3 + 1) ... fails. The real solve here is (3 + 4 + 7) - 2 x 2 ... fails too. Hand 2,3,4,7 is a borderline case that typically yields 3 + 7 x (4 - 2)/2 ... these are the hands worth memorizing by hand rather than scanning.
The notoriously hard four-digit sequences include 1,1,5,8 (the fractional classic 8/(1-1/5) in some rulesets that allow it) and 3,4,7,8 which forces 8 x (3 - 4/7) ... these sit at the edge of the solvable set. A player who memorizes the ten hardest hands has banked the slowest hands in the deck and dropped their average solve time by roughly twenty percent, which matches the pattern seen in the broader Countdown numbers round training literature.
The Digit Sets That Resist A Solve
Across all 10,000 ordered four-digit sequences, exhaustive search finds roughly 8,900 solvable and 1,100 unsolvable. Hands of four zeros, three zeros plus any single digit, and 1,1,1,1 produce a bounded range that never reaches 10. Hands dominated by 1s and 2s cap below 10 under multiplication and above 10 only through sum paths that leave digits unused. The quickest impossibility check is to compute the maximum product of the four digits and confirm it reaches at least 10, then confirm at least one pivot pair is buildable from a two-card subset. Classroom decks and app-based solvers typically pre-screen hands to filter dead sequences, which keeps engagement high and session completion rates near 95 percent.
A practical lookup table for the hundred most-frequent dealt hands fits on a single index card. Build it by sorting your last two hundred solve sessions by solve time, isolating the slowest fifteen percent, and writing out the canonical solution for each. The table is personal because solve speed varies by which pivots a player has drilled, and the slowest hands for a 2 x 5 specialist differ from those of a sum-route specialist. Players who log their results on the Mathness leaderboard tend to converge on similar hard-hand lists within a month.
A Six-Week Drill That Compresses Solve Time
The published mental-math training literature agrees on one pattern: short daily sessions with a hard time cap outperform long weekly sessions by a factor of three. Week one: ten hands per day drawn from the 1-to-6 range, which produces easier pivots. Target solve time ten seconds per hand. Week two: ten hands from 1 to 9, target fifteen seconds. Weeks three through six: timed sprint of twenty hands at ten seconds each with a forced-pass if the timer expires. The forced-pass mechanic trains the scan to abandon dead paths rather than grind through unproductive arithmetic, which is the single largest source of solve-time variance in intermediate solvers.
The All Ten reflex transfers cleanly across adjacent games. The pivot-first scan habit is the same skill rewarded in the Mathness daily round and in the 24 Game, and the signed-pair correction habit is the core of the Countdown numbers round. A player who commits three minutes a day for six weeks has a top-decile reflex across all three games, which is the cheapest return in minutes-per-skill the mental-math space offers. Pair the drill with a longer-form daily game available on the Mathness menu and the two reinforce each other across the week.
