Pentomino solving principles and steps
Here are 3 principles for solving pentominoes, then one free puzzle solved start to finish using them. The rules and controls are in How to play, and answers for the well-known shapes are in Pentomino solutions.
1Solving principles
Start with the piece that has the fewest spots
Pieces differ in how many places they can go. Here are the counts for an empty 6×10 frame, including turns and flips. The fewer places a piece has, the more it helps to settle it early.
| Piece | Placements (6×10) |
|---|---|
| X | 32 answers |
| I | 56 answers |
| T | 128 answers |
| V | 128 answers |
| W | 128 answers |
| Z | 128 answers |
| U | 152 answers |
| L | 248 answers |
| N | 248 answers |
| Y | 248 answers |
| F | 256 answers |
| P | 304 answers |
The cross-shaped X looks the same however you turn or flip it. With only one orientation, it has the fewest. You don't need to count them all. Just pick the piece that looks like it has the fewest places to go.
Keep every gap a multiple of 5
Before placing a piece, check that it won't shut off a gap from the rest. How to solve explains why.
When stuck, back up one piece at a time
Even when the principles narrow things down, you may not get to a single spot. Then pick one and go on. If it turns out the frame can't be filled, take back pieces starting with the last one placed and try another spot. If nothing fits however often you try, go back as far as the pieces you placed first.
Other tips (lay the I along an edge, flip a piece facing the wrong way, and so on) are in the How to solve section of How to play. What all the answers reveal is in Pentomino solutions.
2Solving one puzzle start to finish
We'll solve Normal No.19: put the 5 pieces F, L, P, U and X into a 5×5 frame.
Step 1: Narrow down the X
Count the placements of the 5 pieces in this frame: X has 5, U 36, F 48, L 56 and P 88 (only those that keep to the gap principle). Settle the X first, since it has the fewest.
Put the middle of the X near a corner and a single square is left in that corner, which can never be filled. That leaves 5 spots: the middle of the frame and the middle of each of the 4 edges.
Step 2: Try the middle, then back up
With the X in the middle, there is nowhere left for the F. The F has to be used, so the X can't go here. Take it back.
Step 3: Put the X at the middle of an edge
Turn the frame and the middles of the 4 edges all look the same, so it doesn't matter which you use. Put it at the middle of the top edge.
Step 4: Next fewest, the F
Recount the placements of the remaining pieces: F has 2, U 3, P 4 and L 8. The F has the fewest, and its 2 spots are below the X, on the left and on the right. Those 2 are the same when mirrored left to right, so either will do. Put it lower left.
Step 5: Place the pieces with only one spot
Without leaving a gap, the P and the U each have one spot and the L has 2. Place the P and the L is down to one too. Place the L, and the U goes in last.
Questions
Which piece should I place first?
The one with the fewest places it can go. In a puzzle with the cross-shaped X, that's the X. In an empty 6×10 frame the X has 32 placements, about 1/10 of the 304 for the P, which has the most.
What if I find it can't be filled after placing a piece?
Take back pieces starting with the last one you placed, and try another spot. Backing up isn't failing. It's part of solving.
Is there only one answer?
It depends on the puzzle. The one in this article (Normal No.19) can be filled in 8 ways, but they are all the same shape once turned or flipped. Any way that fills the frame with no gaps is right.