# What is the perimeter of a pentomino which can tile this heart-shaped board?

The puzzle is as follows:

The figure from below shows a board that is made up of 30 squares whose sides are of 2 centimeters each. Such board must be split in six congruent pieces. These must be made by those squares of 2 cm in length. Find the perimeter of one of these pieces which it can be obtained.

The choices given are:

1. 24 centimeters
2. 20 centimeters
3. 18 centimeters
4. 22 centimeters

This puzzle seems to be an adaptation from an APA IQ/psychometric exam from the late 1970s, such has been also referenced in similar IQ test charts from Leon Thurstone and Catell's of the mid 1960s.

I've been trying for more than thirty minutes or so, and I feel I am not getting anywhere. Please, enlighten me.

I can't offer much in the solution. No matter how I have tried to split the grid, some of the pieces overlap. I tried making all sorts of figures with 5 squares each.

As:

$$6\times \frac{\textrm{5 squares }}{\textrm{1 piece}}=30$$ squares

Even my remembered zentris abilities of the late 1990s (in my old 80386DX) have not helped me much. In other words, I've tried making crosses, L's and T's and even S's but none of these have worked.

Is there more than meets the eye here? Perhaps someone with better spatial abilities than me can spot a figure because I can't.

The problem doesn't specifically disallow overlapping the congruent pieces. But I don't think this is reasonable as it indicates that the resulting pieces would come from dissecting the whole board. I just can't make a figure which would fit nicely in all of this.

One thing which I have not tried is dissecting the figure on its diagonals, as I think this solution assumes that the squares must remain intact. But I'm stuck with that.

Can someone help me here?

• There are only 12 shapes to consider. You've already tried X, L, T and Z (Using Golomb's naming). There are only I,P,F,N,W,U, Y, and V left. Work out the shapes by hand, the labels are a good clue. Two of them work, and they have the same perimeter. – theonetruepath Mar 12 at 4:05
• Can we mirror the piece? – justhalf Mar 12 at 4:45
• All pentominos have the same perimeter except for the P pentomino, and that one clearly cannot tile the heart shape. – Jaap Scherphuis Mar 12 at 7:38

As theonetruepath suggested, there are two shapes that work:

The N-shape

and

The F-shape

Both

have perimeter of 12 square sides, or 24 cm.

• I'm sorry for the late reply, but I'm accepting your answer. It is very nice for you to add colors that made me understand it. Great! – Chris Steinbeck Bell Mar 21 at 2:26