Nick's Mathematical Puzzles

Puzzle 160: Absolute maximum

nickh@qbyte.org (Nick Hobson) — Wed, 8 Aug 2007 08:03:16 GMT

The absolute value of a real number is defined as its numerical value without regard for sign. So, for example, abs(2) = abs(-2) = 2. The maximum of two real numbers is defined as the numerically bigger of the two. For example, max(2, -3) = max(2, 2) = 2. Express: (a) abs in terms of max; and (b) max in terms of abs.

Puzzle 159: Eight odd squares

nickh@qbyte.org (Nick Hobson) — Wed, 23 May 2007 14:18:52 GMT

Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares. For example, 6 = 2^2 + 1^2 + 1^2 is the sum of three squares. Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.

Puzzle 158: Fermat squares

nickh@qbyte.org (Nick Hobson) — Thu, 10 May 2007 14:30:59 GMT

By Fermat's Little Theorem, the number x = (2^(p-1) - 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?

Puzzle 157: Trigonometric product

nickh@qbyte.org (Nick Hobson) — Fri, 4 May 2007 22:12:23 GMT

Compute the infinite product [sin(x) cos(x/2)]^(1/2) * [sin(x/2) cos(x/4)]^(1/4) * [sin(x/4) cos(x/8)]^(1/8) * ... , where 0 <= x <= 2*Pi.

Puzzle 156: Three simultaneous equations

nickh@qbyte.org (Nick Hobson) — Fri, 27 Apr 2007 13:02:34 GMT

Find all positive real solutions of the simultaneous equations: x + y^2 + z^3 = 3, y + z^2 + x^3 = 3, z + x^2 + y^3 = 3.