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Tangram Paradoxes

What's a Paradox?

A paradox is typically associated with a verbal or written statement. Basically, it is a statement that doesn't make sense, or one that doesn't support the stated conclusion. A visual paradox is something that seems like it shouldn't or couldn't exist or be possible. Some people refer to optical illusions as paradoxical, such as the classic one that appears on one end to have 3 round rods and the other end is 2 rectangular bars.

Tangram Paradox Optical Illustion

What's a Tangram Paradox?

A Tangram Paradox is a pair of tangrams that have identical silhouettes, except for one small difference, such as an apparent chip in a bowl, or an extra "bump" on a side. They're paradoxes because it doesn't seem possible to build two so similar, but different, images with the same 7 tans. The best known such paradox is called, "The Monk." This is a pair of tangrams of a person, with the only obvious difference being that one image has feet and the other does not.

Tangram-Paradox-the Monk 1 Tangram-Paradox-the Monk 2

In some cases, there can be a series of puzzles that are all paradoxes, as they all form slight deviations of the same, basic shape, such as squares with triangular or rectangular holes in them. Here are some pairs of Tangram Paradoxes:

Tangram-Paradox Square with Triangle Tangram-Paradox Square with parallelogram Tangram-Paradox E Tangram-Paradox E 2

I like this one of the Loch Ness Monster that has an extra wave or bump in the second one.

Tangram-Paradox Nessie 1 Tangram-Paradox Nessie 2

This is a more traditional one of a bowl and a bowl with a nick missing from the bottom.

Tangram-Paradox Bowl 1 Tangram-Paradox Bowl 2

How's It Possible?

There's really nothing mysterious or even missing in paired tangram paradoxes. It's really just a matter of clever redistribution of space. Each individual tan that contributes to a tangram takes up a given amount of space. When all 7 tans are combined, the tangram as a whole takes up a total amount of space or area. That space or area is distributed over the various area that the tangram's shapes cover. Because every tangram uses all 7 tans, every tangram, regardless of its shape, covers the same amount of total area. Some may seem to cover more area because of how wide or tall they are, but as they become wider, they must become shorter, or the white spaces in the tangram image must grow larger, because the images all use the same 7 tans.

When viewing paired paradoxical tangrams, if one appears to be missing a piece somewhere, take a look elsewhere on the tangram. You should be able to see where it has grown fatter or taller in some other area. This may not be very noticeable in drawings of tangrams, such as those on websites, because the drawings may not be perfectly proportional. It should be more noticeable if you physically build the images with tangible tans.

How It's Done

Here's an example of a pair of tangram paradoxes that should be easy to spot where there's a difference between the two, but also where the "extra real estate" came from or is hiding in one or the other. This is a pair of Indians with a bow. Notice that one Indian has a back foot, while the other only has a stub of a leg that represents a leg and foot.

Tangram-Paradox Indian with a bow 1 Tangram-Paradox Indian with a bow 2

Take a look at their bows. You should be able to tell that one Indian's bow is larger than the other's. The area that was taken from the bow has been transferred down to make the foot for the second Indian. This doesn't mean it has literally been shaved off of the tan used for the bow and dumped down there. It just means that a different tan has been used for the two bows and the remaining 6 tans have been rearranged to make roughly the same body image of the Indian. Because the remaining tans used for the second Indian has more total area to them, one of the triangles can be used to form his foot.

Tangram Paradox eBooks

If you like Tangram Paradoxes, you should like these two ebooks. They contain the solutions to the above tangrams, and several more pairs to ponder, plus various tangram activities.
Tangram Paradox Book 1 Tangram Paradox Book 2

Tangram Paradox Fonts

There are also Tangram Paradox Fonts available from the Tangram Fury Font page, along with various other fun fonts. With these fonts, you can place tangrams into your projects just by typing them. The fonts include both the silhouettes to solve, and their solutions. You can get to the Tangram fury Font page below.
Tangram Paradox Font 1

Tangrams Over Time

Over the years, many tangram puzzles have been made, including paradoxes. The most prolific tangram puzzler was Sam Loyd, who published a book called, "The 8th Book of Tan," in 1903. Some of the puzzles he included in his book have never been solved, causing speculation that they have no solutions and are merely hoaxes to confound tangram enthusiasts. Some of those unsolvable puzzles were tangram paradoxes.