Numbers are Playthings

Inhale

Ok. It’s time for a little breather. On 17 March 2024 we began a series of posts on e. Then we moved to another series on principles of counting. Then last week we completed a series on π. All these were quite heavy and I’m sure some of you are still reeling from the shock caused by the posts on π. So for a few weeks I think we should lighten up a bit. I will dedicate the next few posts to recreational mathematics.

From as long as I can remember, I have enjoyed playing with numbers. I enjoyed performing all sorts of operations on them in the attempt to recognize patterns. At an early age, and I believe independent of any external influence, I noticed that when you multiply single digit numbers by nine, the digit in the units place keeps reducing by 1 while the digit in the tens place increases by 1.

Playing with numbers is exhilarating. The joy that one can get from a few simple operations cannot be expressed. In this post, I wish to give you one way I have occupied myself for endless hours and one easy but mathematically based trick. The challenge is to use four copies of the same number (from 1 to 9) and any of the mathematical operations (+, −, ×, ÷, a^b, and √a), with as many sets of parentheses to make as many numbers as possible.

Massaging the Numbers

For example, we can choose to use four ‘2’s. In that case, we can have:

(2 ÷ 2) ÷ (2 ÷ 2) = 1

(2 ÷ 2) + (2 ÷ 2) = 2

(2 + 2 + 2) ÷ 2 = 3

(2 + 2) ÷ 2 × 2 = 4

2 × 2 + 2 ÷ 2 = 5

2 × 2 × 2 – 2 = 6

2^{(2 + 2 ÷ 2)} = 8

(2 + 2 ÷ 2)² = 9

2 × 2 × 2 + 2 = 10

You get the idea. Is there a way of getting 7 as the answer? What about 11? What’s the largest number you can get using four ‘2’s?

We could repeat the same with another number. Try it on your own. Believe me, it keeps your mind engaged and, when you make a breakthrough, there is a serious dopamine rush.

A Constant Result

Now concerning the trick I mentioned earlier. Ask someone to choose a three digit number where the hundreds digit and the units digit as not the same. Now ask them to reverse the order of the digits. So if they choose 356, after reversing, they get 653.

Now ask them to subtract the small from the larger. With the above numbers 653 – 356 = 297. (If they get a two digit number ask them to place a 0 in the hundreds place. So if they had chosen 433, they would have 433 – 344 = 099.) Ask them to reverse the digits again and add the two numbers. So with the original number we chose we would get 297 + 792 = 1089. (If they had 099, they would now get 099 + 990 = 1089.) But tell them not to give you any of these results.

Now, the answer is always 1089. So you could keep a book with around 200 pages and memorize the 9^th line on page 108. So when they have finished adding, tell them to turn to the page indicated by the first three digits (i.e. 108) and check the line indicated by the units digit (i.e. 9). Now you read out the line you have memorized and bamboozle everyone. Of course, since the answer is always 1089, you cannot do this trick with the same person multiple times.

But why does it work? Say we have a in the hundreds place, b in the tens place, and c in the units place. For now let’s assume that a > c. Then the value of the original number is 100a + 10b + c

When we reverse the order, we get a number whose value is 100c + 10b + a. When you subtract this from the original number you will get 100(a – c) + (c – a).

Now since a > c, it followed that c – a is negative. Hence, when we subtracted, we would have had to borrow. However, from the form 100(a – c) + (c – a), there is nothing that represents the tens place. So we will have to borrow 1 from the hundreds place, and then a 1 from the tens place, giving

100(a – c) + (c – a) = 100(a – c – 1) + 10 × 9 + (10 + c – a)

When we reverse this number we get 100(10 + c – a) + 10 × 9 + (a – c – 1)

Now when we add the last two numbers we get

100(a – c – 1 + 10 + c – a) + 10 × 9 + 10 × 9 + (10 + c – a + a – c – 1)

which on simplification gives 900 + 180 + 9 = 1089

Exhale

As we have seen, numbers provide us with a lot of entertainment. We will continue to look at this lighter side of mathematics in the few posts that follow. Don’t forget to try your hand with four ‘3’s or four ‘4’s. And astound someone who does not read this blog with the trick. And tell them to come over here and read the other posts. I’ll see you next week.