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Consider two solids:

  1. a unit-length tetrahedron [triangular base pyramid] and
  2. a unit-length square pyramid.
The visible sides of these solids, if the single square face is placed on the table, are equilateral unit-length triangles.

Glue a face of the tetrahedron to a face of the pyramid so that the points of the two triangles exactly coincide.

How many faces does the new solid have?

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7?

5 (square pyramid) + 4 (tetrahedron) - 2 (faces that stick together) = 7

Amazingly, that's not the answer.

Why?

This is a somewhat infamous puzzle question.

It was given on a national [u.S.] test and the Educational Testing Service listed your answer as correct.

It was challenged by a bright student and ETS was found to be in error.

Later tests were screened by an expert panel before being administered. ;)

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Amazingly, that's not the answer.

Why?

This is a somewhat infamous puzzle question.

It was given on a national [u.S.] test and the Educational Testing Service listed your answer as correct.

It was challenged by a bright student and ETS was found to be in error.

Later tests were screened by an expert panel before being administered. ;)

5 !

I haven't calculated yet, but base on your reply, I feel that the other 2 plane of the tetrahedron are so happen to be co-plane with 2 sides of the square pyramid. Tha't makes 2 surface become 1.

Did I get your message right?

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5 !

I haven't calculated yet, but base on your reply, I feel that the other 2 plane of the tetrahedron are so happen to be co-plane with 2 sides of the square pyramid. Tha't makes 2 surface become 1.

Did I get your message right?

Yup! Nice going. ;)

Doubters can construct the solids fairly easily to verify, but there's a quicker way to see it:

Draw two square pyramids with touching sides of their square bases.

Draw a line connecting their apexes.

Observe that its length is unity, the same as all the other edges.

Now note that the added line defines a unit-length tetrahedron between the pyramids.

A moment's reflection shows that two of the tetrahedron's sides are coplanar with two of the pyramid's sides.

Q.E.D.

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Yup! Nice going. ;)

Doubters can construct the solids fairly easily to verify, but there's a quicker way to see it:

Draw two square pyramids with touching sides of their square bases.

Draw a line connecting their apexes.

Observe that its length is unity, the same as all the other edges.

Now note that the added line defines a unit-length tetrahedron between the pyramids.

A moment's reflection shows that two of the tetrahedron's sides are coplanar with two of the pyramid's sides.

Q.E.D.

Wow, thanks for your explanation. It makes my imagination clearer. And I would say your explanation are so simple and straight forward.

Thank you

This is a nice one! :)

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