Nick's Mathematical Puzzleshttp://www.qbyte.org/puzzles/Mathematical puzzles, with hints, full solutions, and links to related math topics. Fun, with an educational element.enCopyright Nick Hobson, 2002-2007nickh@qbyte.org (Nick Hobson)nickh@qbyte.org (Nick Hobson)Wed, 8 Aug 2007 08:03:16 GMTScience/Math/Recreations/Games_and_Puzzles720Nick's Mathematical Puzzleshttp://www.qbyte.org/pythagoras.gif3232http://www.qbyte.org/puzzles/Wed, 8 Aug 2007 08:03:16 GMTPuzzle 160: Absolute maximumhttp://www.qbyte.org/puzzles/puzzle16.html#p160The 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.nickh@qbyte.org (Nick Hobson)http://www.qbyte.org/puzzles/puzzle16.html#p160Wed, 8 Aug 2007 08:03:16 GMTPuzzle 159: Eight odd squareshttp://www.qbyte.org/puzzles/puzzle16.html#p159Lagrange'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.nickh@qbyte.org (Nick Hobson)http://www.qbyte.org/puzzles/puzzle16.html#p159Wed, 23 May 2007 14:18:52 GMTPuzzle 158: Fermat squareshttp://www.qbyte.org/puzzles/puzzle16.html#p158By 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?nickh@qbyte.org (Nick Hobson)http://www.qbyte.org/puzzles/puzzle16.html#p158Thu, 10 May 2007 14:30:59 GMTPuzzle 157: Trigonometric producthttp://www.qbyte.org/puzzles/puzzle16.html#p157Compute 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.nickh@qbyte.org (Nick Hobson)http://www.qbyte.org/puzzles/puzzle16.html#p157Fri, 4 May 2007 22:12:23 GMTPuzzle 156: Three simultaneous equationshttp://www.qbyte.org/puzzles/puzzle16.html#p156Find 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.nickh@qbyte.org (Nick Hobson)http://www.qbyte.org/puzzles/puzzle16.html#p156Fri, 27 Apr 2007 13:02:34 GMT