Let’s admit it, most of us are intellectually lazy, and thinking is a strange and hard activity, because it requires effort. Even philosophy, the prime discipline devoted to thinking, mostly circulates the ideas of other people, and has some similarity to journalism. Real thinking is hard, and I will always respect Kant for the simple advice he gave to us: Have the courage to think for yourself! To say this differently, philosophy has a justification problem in relation to mathematics, and any self-respecting philosopher ought to be interested in mathematics and logic.
Thankfully, thinking is also addictive, and if we find a problem, we instinctively try to solve it. You add and subtract some numbers, put something in or take it out, and suddenly, here is a true statement that does not seem to depend on real facts any more – how did this happen? There are plenty of problems that can be formulated in simple terms, but they require mathematical formalism in order to solve them, and the insights are often surprising. Here is a problem that is going to make you think on your own if you read on for one or two more minutes:
A magician got tired of the usual magic tricks with hats and rabbits, so he incorporated tricks with numbers into his performance. Here is one of them: He asks a member of the audience to come up with a two-digit number, and then write it down four times in a row, so now she has an eight-digit number. For example, if the original number is 17, the participant writes down 17171717, or 17,171,717. Then the magician asks for the birth date of the audience member. After a short moment of reflection 1 the magician claims that the force is not entirely with him, so he cannot figure out the original eight-digit number, but he now knows one of its divisors, and it is 73! He asks the participant to use a calculator and try it out, and sure enough, he was right! How did he do this?
Certainly, not every eight-digit number (everything between 10 million and 100 million) is dividable by 73. Therefore, the pattern by which the number was created must contain the answer to the problem.
Now is the moment for you to stop reading and start trying it out for yourself.
Here is the solution:
If you write down a two-digit number four times, you can also write the result as the sum of four numbers, and each one is 100 times larger than the original number, since you advance them in the system of decimal placeholders.
You could also write it down in this way:
17,171,717 = 1 x 17 + 100 x 17 + 10,000 x 17 + 1,000,000 x 17
Substituting 17 for a variable a, you get:
1 x a + 100 x a + 10,000 x a + 1,000,000 x a = Eight-digit number
Now you can transform the equation with brackets:
Eight-digit number = a x (1 + 100 + 10,000 + 1,000,000). Now add up what’s in the brackets:
Eight-digit number = a x 1,010,101
In the resulting product, if one factor is dividable by 73, the total number is also dividable by 73. 73 is a prime number, so all you have to do is to find the prime number components of 1,010,101. You do this through a procedure called “prime number factorization,” and there are plenty of online calculators that will give you the results. Sure enough, 1,010,101 = 73 x 13,837, or: 73 x 101 x 137.
Any eight-digit number created through the magician’s procedure is dividable not only by 73, but also by 101, 137, or 13,837!
That’s the beauty of numbers…..
- That mysterious moment of deep thought! ↩