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This puzzle appeared in the 'Skeptical Adversaria' (the newsletter of ASKE, the Association for Skeptical Enquiry), March 2005.
There are four puzzles this month. Puzzles 2, 3 and 4 are each related to their predecessor in some way (in one case rather obscurely).
Two men have challenged each other to a dual with pistols. At the moment they are facing in opposite directions, one due east and one due west. How are they able to see each other without the aid of a reflecting surface?
You are marching in a line of people. How is it possible for the person behind you to be, at the same time, in front of you?
Of all positive whole numbers, half are odd and half are even. Correct?
Think of any number at random. What is the probability that you have chosen any given number, x?
The answers are given below, so do not look until you are ready.
The answers are as follows.
One answer (and possibly the only one) to the first puzzle is that the dualists are facing one another.
The answer to the second puzzle is that the people are marching in a circle.
The third puzzle is only correct for a finite even number of positive integers (e.g. 1 to 100). But, of course, there is an infinite number of such integers. So we have:
1st odd number is 1
2nd odd number is 3
3rd odd number is 5
etc.
There is no limit to the numbers in the first column, likewise the second column. Therefore, the number of odd numbers (and even numbers) is the same as the number of numbers. This is also true of the number of integers divisible by 3, 4, 10, 100, etc, the number of prime numbers, and so on.
I actually dreamt up the fourth puzzle, but I am sure it’s been thoroughly debated already, been the subject of learned papers beyond my understanding, and so on.
By way of introduction, suppose a number (x) is chosen at random from the numbers 1 and 10 inclusive. The probability that x is a certain number, say 5, is of course 1 divided by the number of possible numbers - i.e. 1/10. If x is chosen at random from the numbers 1 to 100, the probability is much smaller, namely 1/100. It’s even smaller for the numbers 1 to 1000, namely 1/1000. So what is the probability when there is no limit to the range of numbers from which x is chosen? Well, the number of numbers is infinite, so the probability that any given number, x, is selected is 1 divided by infinity, which is zero!
Things are no less bizarre when we note the following. Normally x would be twice as likely to be chosen when two different numbers are selected at random. Thus the probability that 5 will be chosen from the numbers 1 to 10 rise from 1/10 to 2/10. The more choices, the greater the probability that a given number will be selected. But with no restriction on the numbers from which to draw from, no matter how many numbers are drawn at random, (2, 10, 100, one million, one billion, etc.) the probability that any given number, x, will be amongst them is still zero.
I am not a mathematician but my guess is that what this all means is that choices cannot be made at random (i.e. all members have the same probability of being drawn) from an infinite set and this is a property of mathematics and not a limitation of the person or device making the selection.
Note: The Infinite Book by John D. Barrow is a curate’s egg of a book that discusses the history, nature and implications of the concept of infinity (publishers Jonathan Cape).
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