Why 2020 is a "magic year"

2020

Let me tell you something: 2020 is a magic year. But what do I mean by that? To explain, grab a calculator and follow these instructions.

  • Think of a two-digit number. Preferably not a round number like 50, or a number with a pattern like 33, just a random (in the everyday sense of the word) two-digit number. But, really, any number will be OK.
  • Now, enter that number into your calculator, and repeat it to create a four-digit number. For example, if you thought of 13 you would now have 1313.
  • As the word "magic" has five letters, multiply your four-digit number by 5.
  • As the word "year" has four letters, multiply your previous answer by 4.
  • You will now have a five- or six-digit number.
  • Divide this number by your original two-digit number, and what do you get?

The result is always 2020.

Here's why this works

Let the thought-of number be n. Assume that this two-digit number is written as AB (so that n = 10A + B). 100n would be written as AB00, and 101n would be written as AB00 + AB = ABAB. That is, transforming AB into ABAB is equivalent to multiplying the original two-digit number by 101.

Multiplying ABAB by 5 gives the answer 101n x 5 = 505n. Multiplying this new number by 4 gives 505n x 4 = 2020n. Dividing by the original number n gives 2020.

The trick will work for any year where the third and fourth digits are the same as the first and second. For example, 1818, 1919, 2020, 2121, 2222, 2323, … .

It would be possible to devise procedures for other years, but they would inevitably be more complicated. The advantage of this trick is that even the most innumerate helper would be able to copy the two digits which they have just entered into a calculator.

To make the procedure work for any four-digit year m written in the form XYXY where m = 101(10X +Y) and is a composite number, two words of lengths j and k, where 10+Y = j.k, would be needed.

If 10+Y is a prime, as for 1919 and 2323, then the trick would not be particularly effective as there could be only one multiplier, and as it would be the first half of the year it would rather give the game away.

The trick would work well in 2828 as the words “magical year” could be used. But it is unlikely that anyone reading this would be around to use it.

2020 is a self-descriptive number

A self-descriptive number is one where, reading its digits from left to right, the first digit gives the frequency of 0s in the number, the second digit gives the frequency of 1s in the number, and more generally the nth digit gives the frequency of (n-1)s.

As an example, Martin Gardner used the idea as a puzzle in Mathematical Circus (page 128) where he asks for a 10-digit number to be written in the 10 cells (below) such that the digit in each cell gives the frequency of the digit above the cell.

0 1 2 3 4 5 6 7 8 9
                   

The answer is 6,210,001,000, which gives:

0 1 2 3 4 5 6 7 8 9
6 2 1 0 0 0 1 0 0 0

That is, 6,210,001,000 contains six 0s, two 1s, one 2, one 6, and no other digits. The answer is unique.

Only six other self-descriptive numbers exist in the base 10 system: 1,210; 2,020; 21,200; 3,211,000; 42,101,000; and 521,001,000.

So, after 2020 it will be a long wait for the next self-descriptive year. The Journal of Recreational Mathematics (Vol 11, 1978-79, pages 76-77) gives a general analysis for number bases other than 10.


About the author

Alan Jackson retired in 2018 and for the previous 27 years was a statistician with the Welsh Government. Before that he worked in the Operational Research Department of British Steel, taught mathematics in a girls’ school, and throughout the 1980s researched in artificial intelligence for Ferranti Computer Systems, at which time he was also Visiting Research Fellow in the Robotics Department of Bristol University. Jackson is an experienced magician and has done mathematical work for a number of well-known magicians, including Derren Brown. The first part of his car registration number is pi.