The Impossible Puzzle: an Extra Puzzle Piece Into a Filled Board

The way this puzzle works is that I have taken a big area of a rectangular space, and chopped it into tiny other shapes which are almost unnoticeably distributed around and in between the pieces of the puzzle.

Thus, the challenge is to insert the big yellow rectangular piece into the seemingly filled-up puzzle board. It involves finding a way of rearranging the puzzle pieces such that all the tiny areas/spaces are gathered back together to form one big rectangular space for our additional rectangular piece to fit in. It leans towards maths a lot but we wouldn't disturb ourselves with the maths.

I have carefully constructed the shapes of the puzzle pieces so that we don't have to fumble around with protractors and precise measurements. Templates for cutting each part are provided in the steps.

NOTE: I reveal the solution to this puzzle in step 8.

Foam board (0.5cm thick)

Exacto knife/ Box cutter knife

Hobby knife

Metal ruler

Glue

Sellotape

Scissors

Pen

Tip: Use a sharp blade.

Using an exacto knife:

1) Use a ruler as a guide when cutting straight lines.

2) Don't apply too much pressure, Just gently pass the blade (at an angle that leans towards the board) about three to four times to get a clean cut.

Using a hobby knife:

1) Preferably, I use the hobby knife when cutting a curve or tight corners.

2) First puncture the board with the hobby knife, not entirely through its thickness and cut along in an up-and-down sawing motion. Pass the hobby knife again now through the entire thickness of the board to completely cut the piece.

The file attached to this step contains the template for the pieces we need. The template is in A4 dimension (210x290mm) already, download the file and print it on an A4 paper.

Cut a rectangular piece slightly bigger than the size of an A4 paper from the foam board and using your sellotape, tape the printed template on the foam board as shown in the third picture of this step.

Cutting the Pieces Out:

NOTE: Use an exacto knife or box cutter with a sharp blade, preferably a new blade, as cutting foam boards can get quickly ugly (jagged edges) when not done right especially with a bad blade.

Now begin cutting each of the puzzle pieces.

You should have eight puzzle pieces, two big triangles, two small triangles and two pairs of "L" shaped pieces.

Keep the pieces, while we move on to make the puzzle board.

Again, I have attached the file containing the template for the part we are making for this step. Download and print it out on an A4 paper.

Cut a rectangular piece from the foam board slightly larger than the A4 paper and tape the template you just printed on it.

Using the exacto knife and ruler, cut out the outer shape on the template (ignore the inner shapes). Remember not to apply too much pressure while cutting to get a clean cut. Use the hobby knife to cut through the curved corners.

Pick out the part you just cut, it's going to be the base of the puzzle board.

This part will be the surface on which we will arrange the puzzle pieces and it has a side extension that will house the extra rectangular piece.

This part of the puzzle board will be laid on top of the base piece, it'll contain all the puzzle pieces. Essentially, it's a frame around the sides to create a shallow depth to arrange our puzzle pieces.

Just as we've done in steps 2 and 3, print the attached file containing the template on an A4 paper and using your sellotape, tape it onto a rectangular foam board slightly larger than the A4 paper.

To begin with, cut out the inner triangle using your exacto knife and ruler guide. Remove the cut portion.

Cut out the inner somewhat 'P' shape to provide a space for the extra rectangular piece to stay within the frame. Use the exacto knife along the straight sides and use the hobby knife to cut the corners, the curves of which make it easier to pick the extra piece with your fingers. Remove the cut portion and you should have your frame just as in picture 5. This is the final piece.

Now unto assembly.

To begin with, apply glue in between the layers and align the puzzle board frame on the board base properly, glue the frame to the base to have the puzzle board complete.

Cut a rectangle with dimensions 3cm by 1.5cm from the foam board. Cut two rectangular pieces of the same size from coloured paper (I'm using a yellow one) and fasten them on both sides of the piece with a sellotape. wrap the sides of the piece with sellotape too. As this piece would be the last piece to fit in the puzzle board and the first to be removed when rearranging, guiding it with sellotape gives it better durability for repeated handling.

Take the extra rectangular piece (yellow) and place it in the side extension while you fit in the other black pieces of the puzzle according to the pictures above.

The big triangles go in first, followed by the small triangles in the opposite corners and then a pair of the "L" piece forming a rectangle to the left side and another pair to the right side.

TRY SOLVING IT: Try to rearrange the black pieces to create a space large enough for the yellow piece to comfortably fit in.

Divide the puzzle into two symmetrical parts across the midline, take out the 'L' pieces from both sides and switch the places of the two triangles on both sides i.e the small triangles move up while the big triangles move to the bottom corners, rearrange each pair of the two "L" shaped pieces on both sides as shown in picture 5 to 7 and fit them into the puzzle board and now we have a big rectangular space for the yellow piece to fit in.

Not a step though.

The area of the "triangular" shape of the entire puzzle, while arranged without a space for the yellow piece is less than the area of the "triangular" shape formed by the entire puzzle while arranged to provide a space for the yellow piece. No maths rule was broken. The trick is this, the sides of the "triangular" shape unlike an actual triangle aren't straight lines, they make a slight angle thus the shape formed is actually a quadrilateral.

When the puzzle is arranged to fill the board without the yellow piece, the sides of the quadrilateral bulge inwards slightly, but when rearranged to provide a space for the yellow piece, the sides of the quadrilateral bulge outward slightly thus the increase in the area of the quadrilateral is what we have as a vacant rectangular space.

In first and second pictures show the unsolved and solved arrangements of the puzzle respectively, an overlap of the two images as in the third picture reveals a gap in the sides of the quadrilateral showing the added area.