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Nick's Mathematical Puzzles: 151 to 160

151. Painted cubes (2 star)

Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black.  The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag.  A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube.  What is the probability that the outside of this cube is completely black?

Hint  -  Answer  -  Solution

152. Totient valence (4 star)

Euler's totient function phi(n) is defined as the number of positive integers not exceeding n that are relatively prime to n, where 1 is counted as being relatively prime to all numbers.  So, for example, phi(20) = 8 because the eight integers 1, 3, 7, 9, 11, 13, 17, and 19 are relatively prime to 20.  The table below shows values of phi(n) for n less than or equal to 20.

n1234567891011121314151617181920
phi(n)112242646410412688166188

Euler's totient valence function v(n) is defined as the number of positive integers k such that phi(k) = n.  For instance, v(8) = 5 because only the five integers k = 15, 16, 20, 24, and 30 are such that phi(k) = 8.  The table below shows values of v(n) for n less than or equal to 16.  (For n not in the table, v(n) = 0.)

nv(n)k such that phi(k) = n
121, 2
233, 4, 6
445, 8, 10, 12
647, 9, 14, 18
8515, 16, 20, 24, 30
10211, 22
12613, 21, 26, 28, 36, 42
16617, 32, 34, 40, 48, 60

Evaluate v(21000).

Hint 1  -  Hint 2  -  Answer  -  Solution

153. Semicircle in a square (1 star)

Find the area of the largest semicircle that can be inscribed in the unit square.

Hint  -  Answer  -  Solution

154. Triangle in a trapezoid (3 star)

In trapezoid¹ ABCD, with sides AB and CD parallel, angleDAB = 6° and angleABC = 42°.   Point X on side AB is such that angleAXD = 78° and angleCXB = 66°.  If AB and CD are 1 inch apart, prove that AD + DX − (BC + CX) = 8 inches.

Trapezoid ABCD, as described above.

(1) A trapezoid is a quadrilateral with at least one pair of parallel sides.  In some countries, such a quadrilateral is known as a trapezium.

Hint  -  Solution

155. Sum of two powers is a square? (2 star)

Is 2n + 3n (where n is an integer) ever the square of a rational number?

Hint  -  Answer  -  Solution

156. Three simultaneous equations (3 star)

Find all positive real solutions of the simultaneous equations:

Hint  -  Answer  -  Solution

157. Trigonometric product (2 star)

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 less than or equal to x less than or equal to 2pi.

Hint  -  Answer  -  Solution

158. Fermat squares (2 star)

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

Hint  -  Answer  -  Solution

159. Eight odd squares (3 star)

Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares.  For example, 6 = 22 + 12 + 12 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.

Hint  -  Solution

160. Absolute maximum (1 star)

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:

  1. abs in terms of max
  2. max in terms of abs
Hint  -  Answer  -  Solution

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Nick Hobson
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Last updated: August 8, 2007