The problem of the repetitious number

This puzzle comes from the book “My Best Mathematical and Logic Puzzles” by the legendary Martin Gardner, author of numerous books of entertaining and puzzling mathematics and, for 25 years, the Mathematical Games column in Scientific American magazine.

In this little parlor trick you ask spectator A to write down any three-digit number, then to copy the number to a new piece of paper, followed by the same digits again, making a six-digit number. Without looking, instruct spectator A to pass the paper to spectator B without revealing the number to you.

Spectator B is to divide the resulting number by 7, and then pass the result to a spectator C, who divides the result by 11. (Tell them not to worry, there will be no remainder!)

Spectator C passes it to yet one final spectator. This last spectator is to divide the result by 13 and then read the resulting quotient. The result will be the original three-digit number.

Problem: Prove that this trick always works regardless of the initially-chosen number.

Hint: The key to solving this problem is finding a simple mathematical statement of it.

Solution

The key to this problem is that writing down a three-digit number and then making it into a six-digit number by appending the same three digits has the same effect as multiplying by 1,001.

So we can ask, is it true that 1001 * x / 7 / 11 / 13 = x?

Here is a solution built with Prooftoys.

The steps. A proof consists of a sequence of true statements connected by inference rules, each producing a new step. You can produce this proof yourself using the proof builder tool, available below.

1. “Starting the proof” – If this (step 1) does not appear automatically, press the button labeled “Clear Work” and confirm in the pop-up.

   In this step there are two assumptions: R x (“x is a real numbeer”), and the claim that 1001 * x / 7 / 11 / 13 = x. The claim statement also appears as the conclusion, following the =>. When you have demonstrated that the main claim is true, it need no longer be assumed, leaving you with just the desired statement.

2. “Simplify” – When you hover the mouse over the step, the proof builder highlights in red the assumption to be removed. In this case you can simply select the whole assumption. Then in the menu choose “simplify ☆”. In this proof, simplification does the whole job, and you are done.

Try it yourself

Can you build the same proof yourself from just the given statement? Here is a workspace you can use to do the experiment.

Seeing details

Prooftoys encourages you to dig down into any step to see details for it. To see the next level of detail, click on the blue text in the description of the step, for example “simplifying”. The display then expands vertically to show you another level of detail. Click on the same text again or on the box labeled “hide” to hide the details again. If there are many levels of detail you can repeat this process on steps in the detailed display.

What’s next?

There are more logical puzzles and examples in the “hands-on” section.

And if some things here interest you, please check out the “about” page for ways to get in touch.