Skip to main content.
RSS What's this?

Nick's Mathematical Puzzles: 61 to 70

61. Two cubes? (2 star)

Let n be an integer.  Can both n + 3 and n2 + 3 be perfect cubes?

Hint  -  Answer  -  Solution

62. Four squares on a quadrilateral (4 star)

Squares are constructed externally on the sides of an arbitrary quadrilateral.

Quadrilateral, with a square constructed on each side, showing perpendicular line segments of equal length.

Show that the line segments joining the centers of opposite squares lie on perpendicular lines and are of equal length.

Hint  -  Solution

63. Cyclic hexagon (4 star)

A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle.  Find the radius of the circle.

Hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11, inscribed in a circle.
Hint 1  -  Hint 2  -  Answer  -  Solution

64. Balls in an urn (1 star)

An urn contains a number of colored balls, with equal numbers of each color.  Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.

How many balls are in the urn?  (Before the extra balls are added.)

Hint  -  Answer  -  Solution

65. Consecutive integer products (2 star)

Show that each of the following equations has no solution in integers x > 0, y > 0, n > 1.

  1. x(x + 1) = yn
  2. x(x + 1)(x + 2) = yn
Hint  -  Solution

66. Quadratic divisibility (3 star)

Show that, if n is an integer, n2 + 11n + 2 is not divisible by 12769.

Hint  -  Solution

67. Random number generator (3 star)

A random number generator generates integers in the range 1...n, where n is a parameter passed into the generator.  The output from the generator is repeatedly passed back in as the input.  If the initial input parameter is one googol (10100), find, to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1.  That is, what is the expected value of x, after running the following pseudo-code? 

n = 10100
x = 0
do while (n > 1)
n = random(n) // Generates random integer in the range 1...n
x = x + 1
end-do
Hint 1  -  Hint 2  -  Answer  -  Solution

68. Difference of powers (3 star)

Find all ordered pairs (a,b) of positive integers such that |3a − 2b| = 1.

Hint  -  Answer  -  Solution

69. Combinatorial sum (3 star)

Find a closed form expression for  the sum from k = 1 to n of C(n,k) * k^5

Hint  -  Answer  -  Solution

70. One degree (2 star)

Show that cos 1°, sin 1°, and tan 1° are irrational numbers.

Hint  -  Solution

Back to top


Nick Hobson
nickh@qbyte.org
Email Nick.
Last updated: September 17, 2003