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

Euler's totient function (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, (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 (n) for n 20.

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(n) | 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 | 8 |

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

n | v(n) | k such that (k) = n |
---|---|---|

1 | 2 | 1, 2 |

2 | 3 | 3, 4, 6 |

4 | 4 | 5, 8, 10, 12 |

6 | 4 | 7, 9, 14, 18 |

8 | 5 | 15, 16, 20, 24, 30 |

10 | 2 | 11, 22 |

12 | 6 | 13, 21, 26, 28, 36, 42 |

16 | 6 | 17, 32, 34, 40, 48, 60 |

Evaluate v(2^{1000}).

Hint 1 - Hint 2 - Answer - Solution

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

Hint - Answer - Solution

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

(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

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

Hint - Answer - Solution

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

Hint - Answer - Solution

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.

Hint - Answer - Solution

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?

Hint - Answer - Solution

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.

Hint - Solution

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:

- abs in terms of max
- max in terms of abs

Hint - Answer - Solution

Nick Hobson

nickh@qbyte.org