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Nick's Mathematical Puzzles: 31 to 40

31. Area of a rhombus (1 star)

A rhombus, ABCD, has sides of length 10.  A circle with center A passes through C (the opposite vertex.)  Likewise, a circle with center B passes through D.  If the two circles are tangent to each other, what is the area of the rhombus?

Hint  -  Answer  -  Solution

32. Differentiation conundrum (2 star)

The derivative of x2, with respect to x, is 2x.  However, suppose we write x2 as the sum of x x's, and then take the derivative:

Let f(x) = x + x + ... + x  (x times)

Then f'(x) = d/dx[x + x + ... + x]  (x times)
  = d/dx[x] + d/dx[x] + ... + d/dx[x]  (x times)
  = 1 + 1 + ... + 1  (x times)
  = x

This argument appears to show that the derivative of x2, with respect to x, is actually x.  Where is the fallacy?

Hint  -  Solution

33. Harmonic sum (2 star)

Let H0 = 0 and Hn = 1/1 + 1/2 + ... + 1/n.
Show that, for n > 0, Hn = 1 + (H0 + H1 + ... + Hn−1)/n.
(That is, show that Hn is one greater than the arithmetic mean of the n preceding values, H0 to Hn−1.)

Hint  -  Solution

34. Harmonic sum 2 (2 star)

Let Hn = 1/1 + 1/2 + ... + 1/n.
Show that, for n > 1, Hn is not an integer.

Hint  -  Solution

35. Cuboids (2 star)

An a × b × c cuboid is constructed out of abc identical unit cubes -- a la Rubik's Cube.  Divide the cubes into two mutually exclusive types.  External cubes are those that constitute the faces of the cuboid; internal cubes are completely enclosed.  For example, the cuboid below has 74 external and 10 internal cubes.

Example cuboid: 3 by 4 by 7

Find all cuboids such that the number of external cubes equals the number of internal cubes.  (That is, give the dimensions of all such cuboids.)

Hint  -  Answer  -  Solution

36. Composite numbers (3 star)

Take any positive composite integer, m.
We have m = ab = cd, where ab and cd are distinct factorizations, and a, b, c, d greater than or equal to 1.
Show that an + bn + cn + dn is composite, for all integers n greater than or equal to 0.

Hint  -  Solution

37. Five marbles (2 star)

Five marbles of various sizes are placed in a conical funnel.  Each marble is in contact with the adjacent marble(s).  Also, each marble is in contact all around the funnel wall.

Cross section of five marbles in a conical funnel

The smallest marble has a radius of 8mm.  The largest marble has a radius of 18mm.  What is the radius of the middle marble?

Hint  -  Answer  -  Solution

38. Twelve marbles (3 star)

A boy has four red marbles and eight blue marbles.  He arranges his twelve marbles randomly, in a ring.  What is the probability that no two red marbles are adjacent?

Answer  -  Solution

39. Prime or composite? (3 star)

Is the number 2438100000001 prime or composite?  No calculators or computers allowed!

Hint  -  Answer  -  Solution

40. No consecutive heads (3 star)

A fair coin is tossed n times.  What is the probability that no two consecutive heads appear?

Hint  -  Answer  -  Solution

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Nick Hobson
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Last updated: April 30, 2005