Puzzle L1 (three numbers)
Move exactly one match to make the equation true.
Tap a match to pick it up, then tap a dashed spot to put it down. The board checks the math straight away, and it accepts any true equation, not only the one in the key. Hint goes in three steps: where the match comes from, which match it is, and where it lands. With a keyboard, Tab to a match and press Enter, then Tab to a spot and press Enter again.
New to these? Start with the 30 easier matchstick puzzles first. They use the same rules and teach the digit changes these ones lean on.
The rules
- Move exactly one match. No adding, no removing, no breaking, no diagonal matches.
- Digits look like a digital clock. A 1 is two upright matches, a 7 is a 1 with a match on top.
- Signs are matches too. A minus is one flat match, a plus is a flat one and an upright one, an equals sign is two flat ones.
- The result must be a real equation: numbers joined by plus and minus, exactly one equals sign, both sides equal, and no number that starts with a zero.
What makes these hard
A puzzle isn't hard because the numbers are big. It's hard because so many moves look right. Every answer below says how many single moves turn the puzzle into a proper equation that's still false. Those are the dead ends.
On the easier page there are 3 to 13. Here there are between 38 and 158, and exactly one move that works.
The set comes in three kinds:
- Three numbers. Two signs on one side means the match can go to a sign you weren't looking at.
- Sign swaps. The fix moves the equals sign itself, so the two sides of the equation change places.
- Big numbers. Four digit answers full of 8s and 9s, where almost every match can move somewhere.
How I picked them
I didn't write these by hand. A small program started from thousands of true equations, undid one match move in every possible way, and kept each false equation it got. Then the same engine that runs the board above tried every single move on each one and kept only those with exactly one true result. That left a little over 56,000 puzzles.
From those I took the ones with the most dead ends, preferred matches that travel a long way, and allowed each kind of move only once, so you won't solve two puzzles with the same trick. Anything already in my app or on the easier page was left out. The site also checks every answer again on each update with a second solver written separately, just to be safe.
How to crack the hard ones
Staring doesn't work on these. Counting does. Here's the order I'd go in.
1. Work out the gap
Add up both sides as they stand and write down the difference. Then think about what a single move can do to it. A 9 that gives up a match becomes a 3 or a 5, so it drops by 6 or 4. An 8 that loses its middle match becomes a 0. Each of those changes is worth a fixed amount, and only a few of them fit the gap.
2. Flip a minus in your head
Turning a minus into a plus adds twice the number after it. That's a big jump, and it's the move most people miss in long sums.
Take L8, 59 = 16 + 18 − 19. The right side is 15. Make that minus a plus and it's 16 + 18 + 19, which is 53. Now the left side is 6 too big, and a 9 that gives up its upper left match turns into a 3. That's exactly the match the plus needed.
3. Read the equals sign somewhere else
When nothing adds up, take one match off the equals sign and lay it on a minus. The equals sign becomes a minus, the minus becomes an equals sign, and the equation splits in a new place. In S1, 17 − 15 + 10 = 8 is wrong, but 17 = 15 + 10 − 8 is right. Try reading the equation with the equals sign at each minus before you move a single match.
4. Big numbers: guess the answer first
With four digit numbers, don't hunt for the match. Do the sum. In B6, 1993 = 995 − 958, the right side is 37, which is nowhere near. With a plus it's 1953, and 1993 is only one digit away from that. Now there's just one question left: which match turns a 9 into a 5 and also gives the minus its upright match.
5. Use the donor list
Only a few digits can lose a match and still be a digit: 7 becomes 1, 8 becomes 0, 6 or 9, 9 becomes 3 or 5, and 6 becomes 5. Find every one of those in the puzzle. In every puzzle here except the sign swaps, the match comes from one of them, which takes you from dozens of options to a few.
Three numbers
Two signs on one side, sometimes on the right of the equals sign. Watch the sign you're not looking at.
L1
Answer to L1
35 = 19 + 6 + 10. Take the upper left match from the first 9 and put it on the lower left side of the second 5. The first 9 becomes a 3 and the second 5 becomes a 6.
Dead ends: 56
L2
Answer to L2
18 + 13 + 5 = 36. Take the upper left match from the second 9 and put it on the lower left side of the first 9. The second 9 becomes a 3 and the first 9 becomes an 8.
Dead ends: 58
L3
Answer to L3
10 = 8 + 18 − 16. Take the middle match from the first 8 and put it on the lower left side of the 5. The first 8 becomes a 0 and the 5 becomes a 6.
Dead ends: 61
L4
Answer to L4
10 + 16 − 8 = 18. Take the middle match from the first 8 and put it on the lower left side of the 9. The first 8 becomes a 0 and the 9 becomes an 8.
Dead ends: 61
L5
Answer to L5
19 = 9 − 9 + 19. Take the lower left match from the 8 and put it on the second minus. The 8 becomes a 9 and the second minus becomes a plus.
Dead ends: 64
L6
Answer to L6
16 + 18 − 15 = 19. Take the top match from the 7 and put it on the first minus. The 7 becomes a 1 and the first minus becomes a plus.
Dead ends: 66
L7
Answer to L7
19 + 19 + 12 = 50. Take the middle match from the 8 and put it on the minus. The 8 becomes a 0 and the minus becomes a plus.
Dead ends: 66
L8
Answer to L8
53 = 16 + 18 + 19. Take the upper left match from the first 9 and put it on the minus. The first 9 becomes a 3 and the minus becomes a plus.
Dead ends: 83
Sign swaps
All four are fixed by moving a match off the equals sign. The hard part is seeing which minus it goes to.
S1
Answer to S1
17 = 15 + 10 − 8. Take one match from the equals sign and put it on the minus. The equals sign becomes a minus and the minus becomes an equals sign.
Dead ends: 38
S2
Answer to S2
23 = 19 + 18 − 14. Take one match from the equals sign and put it on the minus. The equals sign becomes a minus and the minus becomes an equals sign.
Dead ends: 39
S3
Answer to S3
17 = 19 + 8 − 10. Take one match from the equals sign and put it on the minus. The equals sign becomes a minus and the minus becomes an equals sign.
Dead ends: 44
S4
Answer to S4
6 = 16 + 9 − 19. Take one match from the equals sign and put it on the minus. The equals sign becomes a minus and the minus becomes an equals sign.
Dead ends: 51
Big numbers
Three and four digit numbers with the most dead ends in the set. Do the sum before you touch a match.
B1
Answer to B1
1561 = 968 + 593. Take the upper right match from the first 9 and put it on the lower left side of the third 9. The first 9 becomes a 5 and the third 9 becomes an 8.
Dead ends: 124
B2
Answer to B2
1693 = 997 + 696. Take the upper right match from the 8 and put it on the minus. The 8 becomes a 6 and the minus becomes a plus.
Dead ends: 128
B3
Answer to B3
1851 = 998 + 853. Take the upper left match from the second 9 and put it on the upper right side of the second 5. The second 9 becomes a 3 and the second 5 becomes a 9.
Dead ends: 132
B4
Answer to B4
959 + 989 = 1948. Take the upper right match from the second 9 and put it on the upper right side of the 5. The second 9 becomes a 5 and the 5 becomes a 9.
Dead ends: 134
B5
Answer to B5
1395 = 998 + 397. Take the upper left match from the first 9 and put it on top of the second 1. The first 9 becomes a 3 and the second 1 becomes a 7.
Dead ends: 141
B6
Answer to B6
1953 = 995 + 958. Take the upper right match from the second 9 and put it on the minus. The second 9 becomes a 5 and the minus becomes a plus.
Dead ends: 158
Print them
Printing this page gives you the eighteen puzzles on their own, without answers, and the answer key on a separate sheet. Choose Save as PDF in the print window if you'd rather keep a file.
| Puzzle | Matches show | Answer |
|---|---|---|
| L1 | 95 = 19 + 5 + 10 | 35 = 19 + 6 + 10 |
| L2 | 19 + 13 + 5 = 96 | 18 + 13 + 5 = 36 |
| L3 | 18 = 8 + 18 − 15 | 10 = 8 + 18 − 16 |
| L4 | 18 + 16 − 8 = 19 | 10 + 16 − 8 = 18 |
| L5 | 18 = 9 − 9 − 19 | 19 = 9 − 9 + 19 |
| L6 | 16 − 18 − 15 = 79 | 16 + 18 − 15 = 19 |
| L7 | 19 − 19 + 12 = 58 | 19 + 19 + 12 = 50 |
| L8 | 59 = 16 + 18 − 19 | 53 = 16 + 18 + 19 |
| S1 | 17 − 15 + 10 = 8 | 17 = 15 + 10 − 8 |
| S2 | 23 − 19 + 18 = 14 | 23 = 19 + 18 − 14 |
| S3 | 17 − 19 + 8 = 10 | 17 = 19 + 8 − 10 |
| S4 | 6 − 16 + 9 = 19 | 6 = 16 + 9 − 19 |
| B1 | 1961 = 969 + 593 | 1561 = 968 + 593 |
| B2 | 1893 = 997 − 696 | 1693 = 997 + 696 |
| B3 | 1851 = 598 + 859 | 1851 = 998 + 853 |
| B4 | 999 + 989 = 1548 | 959 + 989 = 1948 |
| B5 | 1995 = 998 + 391 | 1395 = 998 + 397 |
| B6 | 1993 = 995 − 958 | 1953 = 995 + 958 |
Questions
Are these really the hardest matchstick puzzles?
They're the hardest I could find under clear rules. Out of a little over 56,000 one move puzzles with a single answer, these have the most moves that look right but aren't. Hard is personal, though. If you have one that beats these, send it to me.
Is there only one answer to each?
Yes, under the rules above. A program tried every possible move on every puzzle and found exactly one true equation each time. If you allow a not equal sign or a match laid diagonally, you can find others, but those break the rules.
Why do some answers have four digit numbers?
Because the numbers had room for it. 1953 = 995 + 958 is just as true as 3 + 4 = 7, and big sums full of 8s and 9s give the most places to put a match.
Can the equals sign really move?
Yes. An equals sign is two matches, so taking one leaves a minus, and putting it on a minus makes a new equals sign. It's one match moved, which is all the rules ask for.
Is there an app with more?
Yes. My Matchstick Puzzle app for Android has 550 levels, including two match moves, add or remove puzzles and shapes, plus a daily puzzle. It follows the same rules and works offline.
If you get stuck, go back to the gap. Work out how far off the equation is, and the match usually shows itself.
Matchstick Puzzle
550 levels in four kinds of puzzles, a daily puzzle and hints that show which match to move.