THE RULES
You're given six source numbers (a mix of small numbers 1–10 and large numbers 25, 50, 75, 100) and a 3-digit target between 100 and 999. Use any subset of the source numbers, each at most once, with +, −, ×, and ÷ to reach the target. All intermediate results must be whole numbers and positive. You don't have to use all six numbers.
Scoring: exact = maximum points. Within 10 = partial credit. More than 10 away = nothing. Exact is always the goal, but a near-miss beats zero.
START WITH THE LARGE NUMBERS
Large numbers (25, 50, 75, 100) do the heavy lifting. The fastest route to most targets is: multiply or use a large number to get in the right range, then adjust with small numbers.
The most useful multiplication to know: if you have 75 and a 4, 75 × 4 = 300. If you have 100 and a 9, 100 × 9 = 900. These single-step jumps get you into target range immediately — then a small addition or subtraction closes the gap.
Habit: before anything else, check what each large number multiplied by each small number produces. If any product is within 10 of the target, you probably have a solution in two steps.
THE 75 TRICK AND FRIENDLY MULTIPLES
Some targets have a structure that experienced players spot immediately:
- Target divisible by 25: can you reach it by multiplying 25 by a small number you have? 725 = 25 × 29 = 25 × (30 − 1) — if you have 25, 6, 5, and 1, that's 25 × (6 × 5 − 1) = 25 × 29.
- Target close to a round multiple: 496 = 500 − 4 = 100 × 5 − 4. If you have 100, 5, and 4, you're done in two steps.
- Target with a large-number factor: 672 = 8 × 84 — but 84 = 75 + 9. Look for whether the target has a factor that's reachable from your large numbers.
These patterns aren't exhaustive — they're the most common ones. Noticing one saves the 20 seconds you'd otherwise spend searching randomly.
WHEN TO APPROXIMATE
If you can't find an exact solution in the first 15–20 seconds, switch to finding a near-miss within 10. A score is better than zero.
The fastest near-miss strategy: find the closest product of a large number and a small number, then see how many small numbers away from the target you end up. 8 away with numbers 1–5 available is usually solvable. 11 away might not be.
In competitive settings, locking in a near-miss early is often correct — your opponent might not find exact either, and a 7-point declaration beats a 0-point "I couldn't get it."
AVOID EXHAUSTIVE SEARCH
The number of ways to combine six numbers with four operations is large. Exhaustive search — trying every combination mentally — is nearly impossible in 30 seconds and usually unnecessary.
Instead: pattern first, then targeted search. Work through the large-number products, check round-number proximity, and look for the target's structure (is it near a multiple of 25? 50? 100?). This covers 80% of targets. The remaining 20% need creative intermediate steps — two operations before a third — but even there, start from the large numbers and work inward.
THE BEGINNER MISTAKE
Beginners start from the small numbers and try to build up to the target additively. Adding 3 + 8 + 7 + ... eventually reaches most targets — but it takes far too long and misses the fast routes.
The correct mental order is top-down: start with the largest number you have, see what it gets you when multiplied or combined with available numbers, then adjust the remainder with small numbers. Large → medium → fine-tune, not small → small → small → hope.