A puzzle is made of pieces with tabs and slots that join tog

A puzzle is made of pieces with tabs and slots that join together to make a whole. Assume each piece is square and any tab can t into any slot. The gure below illustrates the shapes of the puzzle pieces.

The corner pieces are identical with one tab and one slot; see the left picture.

The edge pieces have a slot on the left and a tab on the right. Half of the edge pieces have a slot on the bottom and half have a tab; see the two middle pictures.

The middle pieces have two slots across from each other and two tabs across from each other; see the right picture.

(a) How many ways are there to put together a 2 2 square puzzle?

(b) How many ways are there to put together a 3 3 square puzzle?

(c) How many ways are there to put together a 5 5 square puzzle?

Hint: Start with the corner pieces, then move on to the edge pieces, and nally think about the

middle pieces.

Each piece is unique.

Rotation of the entire puzzle will be considered as a different way of putting together the pieces. Imagine you are looking at the puzzle, it\'ll be a different picture after it\'s rotated 90 or 180 or 270 degrees.

Solution

1)

For a 2 x 2 puzzle, we have 4 corner pieces only.

The four corner pieces can be arranged amongst themselves in 4! way = 24 ways

2)

For a 3 x 3 puzzle,

we have 4 corner pieces, 4 edge pieces (2 with top slots and two with bottom slots) and 1 centre piece.

Thic can be arranged in 4! * 2! * 2! * 1! ways = 24 * 2 * 2 * 1 = 96 ways

3)

For a 5 x 5 puzzle,

We have 4 corner pieces, 12 edge pieces (6 with top slots and 6 with bottom slots) and 9 center pieces.

Thus,

Number of ways = 4! * 6! * 6! * 9!

= 4.5148 * 10¹² ways

Hope this helps.