Earlier today I set you three puzzles concerning the number 2021, which is the concatenation of two consecutive integers, 20 and 21. Before we get to the problems (and the answers), thanks to reader ConradKnightSocks for alerting me to the brilliant fact that 2021 is also the product of two consecutive prime numbers: 43 x 47.

The last time this was the case was 1763, which is 41 x 43.

The next time this will happen is 2491, which is 47 x 53.

If, however, you want to find out the next date which is *both *a concatenation of consecutive integers *and* a product of consecutive primes, you’re probably going to have to wait for the heat death of the universe (or thereabouts).

**1. Countdown conundrum**

Fill in the blanks in the following equation so it makes arithmetical sense:

10 9 8 7 6 5 4 3 2 1 = 2021

**Solution **Here’s one that works nicely:

10 + (9 + 8 + 7 + 6 )(5 + 4^{3 }– 2) + 1

(Sent in by legendary Scottish maths teacher Chris Smith)

**2. Inder’s enigmas**

Inder J Taneja, a retired maths professor from Brazil, likes to find ways to describe the new date using combinations of the same digit. For example, here’s how he does it with 1.

(1+1)^{11 }−(1+1+1)^{(1+1+1)} = 2021

Can you find combinations of 2’s, 3’s, 4’s 5’s, 6’s, 7s, 8’s and 9’s that also equal 2021? You are only allowed to use at most ten digits per equation. You can concatenate, i.e, place two or more digits together in the same number, as above with 11.

**Solution **Here are Inder’s examples:

(2 x 22 + (2/2))^{2} – 2 – 2 = 2021

(3 – (3/3))^(33/3) – 3^{3}

4 + (4 + 4) x (4^{4} – 4) + 4/4

5 + (((5 + 5)^(5 – (5/5)) + 55)/5) + 5

6 + 66 x (6 x 6 – 6) + 6 x 6 – (6/6)

((77 + 7)/7 )+ 7 x (7 x (7 x 7 – 7) – 7)

8 x (8 +8) x ( 8 + 8) – 8 – 8 – (88/8)

((9 + 9)/9)^(99/9) – (9 + 9 + 9)

(The symbol ‘^’ means ‘to the power of’, thus 2^2 is the same as 2^{2})

Inder has, in fact, written a 75-page article, 21 Mathematical Highlights for 2021, which includes dozens of amazing numerical games involving the number 2012, including palindromes, reverse and upside-down writing, magic squares and pictorial representations. Do check it out!

**3. Marek’s mindbender**

The Polish puzzle creator Marek Penszko sent in the grid puzzle below, which you can print out here.

You must fill in each of the nine empty cells so that all the equations are correct. Each cell requires a single digit. Since 0, 1 and 2 are already visible, the only digits that can appear in the empty cells are the digits from 3 to 9, some of which appear more than once. The calculations are to be done strictly from left to right, or from top to bottom (ie. ignore operator precedence).

**Solution**

20 = 7 + 4 + 9

2 = 4 x 3 : 6

0 = 8 – 5 – 3

I hope you enjoyed today’s puzzle fun. See you all in 2021!

*I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book.*