Problems of the Week

2018-04-23 Basic


Every green square is one quarter of the square it's inscribed in.

If the pattern continues infinitely, what fraction of the total figure will be green?

Why does a full glass of water cast a darker shadow than an empty glass?

A blue light shines on a yellow shirt in an otherwise dark room.

What color does the shirt appear?

A semiprime is a number that is the product of exactly two (not necessarily distinct) primes. Examples of semiprimes are 6=2×3,6=2 \times 3, 35=5×7,35=5 \times 7, and 121=11×11121=11 \times 11.

It's easy to find pairs {21,22}\{21,22\} and triplets {85,86,87}\{85,86,87\} of consecutive semiprimes on this list of semiprimes up to 187.

Is there a run of consecutive semiprimes longer than three numbers?

Is it possible to cut a circle into pieces with these conditions?

  • There are finitely many pieces.
  • The pieces must be congruent, meaning that they have the same shape and area.
  • Some of the pieces have at most one point on the circumference of the circle.
  • Curved cuts are allowed.

This cutting is only <strong>partially</strong> correct. It satisfies all but the "same shape" condition. This cutting is only partially correct. It satisfies all but the "same shape" condition.


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