Some are left as puzzles for you to solve with a button to show the solutions.

Many of the facts below are derived from Neil Sloane's On-line Encyclopedia of Integer Sequences where links to the sequences containing 2021 listed here are shown as, for instance,

A000045 ( the Fibonacci numbers).

Contents of this page

The icon means there is a
You Do The Maths section of questions to start your own investigations.

One inadmissible version from Chris Smith is

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

which is inadmissible because it uses the "to-the-power-of' operation in 4- Using only those 4 arithmetic operations and brackets, can you make 2021 by inserting them in between the
numbers 10 9 8 7 6 5 4 3 2 1?

Show answers10 − 9 × 8 × 7 × 6 + 5 − 4 × 3 × 2 − 1 - What if you were allowed to use the same 10 numbers but in any order?

The next year like this is 2122, in 101 year's time.

A001704

2021 is also composed just of the digits 0,1 and 2.

- It is easy to write the series of numbers which are just 2 successive numbers joined together such as 2021.

Suppose we allowed any number of consecutive integers to be joined together, such as 5678 and 345.

Write down the full list of such numbers in order up to 2021.

How many are there?

Show the answerThe list is:

12, 23, 34, 45, 56, 67, 78, 89, 123, 234,

345, 456, 567, 678, 789, 910, 1011, 1112, 1213, 1234,

1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021

so there are 27 up to 2021.

A035333 - Up to 2021, how many years are there whose largest digit is 2?

Show the answer2, 12, 20, 21, 22, 102, 112, 120, 121, 122,

200, 201, 202, 210, 211, 212, 220, 221, 222, 1002,

1012, 1020, 1021, 1022, 1102, 1112, 1120, 1121, 1122, 1200,

1201, 1202, 1210, 1211, 1212, 1220, 1221, 1222, 2000, 2001,

2002, 2010, 2011, 2012, 2020, 2021, ...

There are 45 such numbers less than 2021, 2021 is the 46th.

A277964 - Suppose now that we want all the numbers up to 2021 that we can write using some of all of the digits 0,1 and 2.

There is a quick way to find out how many are less than 2021. How many are there and what is the quick method?

Show the answerSuch numbers are just the integers written*in base 3*.

If the first is 0, the second 1 and so on then 2021 as a base 3 number is2×3^{3}+ 0×3^{2}+ 2×3^{1}+ 1 = 61

2021 is the 62^{nd}number.

The reverse of 2021 is 1202 whose square is 1444804, which is just the reverse of 2021

Such numbers are not very common ...

- How many numbers can you find whose squares are reversals of each other, up to 1202 (the smaller of 2021 and its reversal)?

Show the answer12, 13, 33, 102, 103, 112, 113, 122, 1002, 1003,

1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103, 1112, 1113,

1121, 1122, 1202, ...

There are 23 including 1202.

A106323

2021 = 43×46 + 43

Also 2021 times its reversal 1202 is the palindromic number 2429242.

A106323

- How many other numbers smaller than 2021 can you find that make a palindrome when they are multiplied by their
reversal?

For example, 211×112 is 23632, a palindrome but this is not the smallest example. In your list include both the number and its reversal, so 112 and 211 in this example. Show the answerThe complete list up to 2021 is:

1, 2, 3, 11, 12, 21, 22, 101, 102, 111,

112, 121, 122, 201, 202, 211, 212, 221, 1001, 1002,

1011, 1012, 1021, 1022, 1101, 1102, 1111, 1112, 1121, 1201,

1202, 1211, 2001, 2002, 2011, 2012, 2021, ...

There are 36 smaller numbers.

A062936

# | 1 | . | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

π = | 3 | . | 1 | 4 | 1 | 5 | 9 | 2 | 6 | 5 | 3 | 5 | 8 | 9 | 7 | 9 | 3 | 2 | 3 | 8 | 5 |

e = | 2 | . | 7 | 1 | 8 | 2 | 8 | 1 | 8 | 2 | 8 | 4 | 5 | 9 | 0 | 4 | 5 | 2 | 3 | 5 | 4 |

φ = | 1 | . | 6 | 1 | 8 | 0 | 3 | 3 | 9 | 8 | 8 | 7 | 4 | 9 | 8 | 9 | 4 | 8 | 4 | 8 | 2 |

There are only 19 other places where all three decimal expansions agree in the first 2020 digits and then at the 2021

A266002

If we try to include square-roots also, the smallest is √46 which has a **3** at place 2021.

2021 = 20 + 21 + ... + 65 + 66

2021 = 26 + 27 + ... + 67 + 68

2021 = 1010 + 1011

The sum of all the numbers up to 2021 is the runsum from 1 to 2021 and is 2043231.

2021 is also the link between two runsums, making the "friendly runsum":

195 + 196 + ... + 2020 + **2021** = **2021** + 2022 + ... + 2850 + 2851 = 2024316

A006094

- When was the last year that was a semi-prime number?
Show the answer
2019 = 3 × 673
- Find all the bases up to base 10 in which 2021 is a prime number.

Show the answer2021_{3}= 61

2021_{4}= 137

2021_{7}= 701

are the only primes.

Once we have one such triangle, we can always multiply all the sides by 2 say to get 6-8-10 or by 3 to find 9-12-15 and so on. All these have the same

If we want

There are 4 Pythagorean triangles with a side of length 2021:

43×(47,1104,1105) = 2021, 47472, 47515;

47×(43,924,925) = 2021, 43428, 43475;

2021,2042220,2042221;

180,2021,2029

No Pythagorean triangle has a hypotenuse (longest side) of 2021.47×(43,924,925) = 2021, 43428, 43475;

2021,2042220,2042221;

180,2021,2029

No Pythagorean triangle has a perimeter (sum of all three sides) of 2021.

No Pythagorean triangle has an area of 2021.

A009000: the hypotenuses

A009096: the perimeters

A055112: the areas

2021! has 5805 digits. It starts 780301939..... but how does it end?

With a 0 because it contains 10 as a factor.

But 100 and 1000 are also factors so that gives at least 1+2+3=6 zeroes at the end.

In fact there are many more zeroes at the end of 2021! .

How many zeroes are at the end of 2021! ?

The answer is in the You Do The Maths... section below.
d= | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

r= | 1 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 1 |

When is the next? See You Do The Maths... section below.

2021 is close to 21

- Find an easy way to compute the number of 0s at the end of 2021 .

Show an answerEach 0 corresponds to a multiple of 10 in the product.

10 = 5 × 2.

There are far more even numbers in the product than multiples of 5, so we can always match each 0 at the end with a factor 5 (and a factor 2) in the product.

2021/5 = 404.2 so there are 404 numbers with a factor 5 2021/5 ^{2}= 80.84 so there are 80 numbers with an extra 5 as a factor 2021/5 ^{3}= 16.168 so there are 16 with another 5 as factor 2021/5 ^{4}= 3.2336 so there are just 5 with one more factor 5 total = 503 ^{5}is 3125 which is bigger than 2021 so that accounts for all the 5 factors in 2021! - What formula will give the number of digits in 2021! ?

Show an answerLet x be Log10 ( 2021! ) then 10^{x}is 2021! .

When rounded up, this gives the number of decimal digits in 2021! (since this is not an exact power of 10)

(Log10[101] is 2.00432 and Log10[999] is 2.99957. All the values of the logs to base 10 of the numbers from 100 to 999 begin with 2 and all contain 3 digits.Find out more about logs and how to use them on the Formula for Fibonacci Numbers page

- Which was the first year to have a remainder of 5 when divided by 6 and 7 and 8 and 9? (Easy!)

How does the list continue up to 2021? (Harder)

When is the next year with this property?

Show an answerThe first year was year 5.

If we add 6 to any year with remainder 5 when divided by 6, then it also will have the same remainder (mod 6).

If we want multiples of both 6 and 7 to have remainder 5, we can add 6×7 = 42.

The Lowest Common Multiple of 5, 6, 7 and 8 is 504.

Any multiple of 504 can be added to 5 to produce a number with remainder 5 when divided by each of 6, 7, 8 and 9:

The list continues... 5+504=509, 5+2×504=1013, 1517, 2021, 2525 - Make a list of all the whole numbers that are a
**perfect power**, that is, are of the form a^{b}for whole numbers a and b, up to 400.

Show the answerUp to 400 they are:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400

A001597 - How do you find numbers close to 2021 of the form b
^{p}with the power p is a "simple" fraction?

Show an answerIf b^{p}= n then log_{b}(n)= p, "the log to base b of n is p".

For "simple" powers which are whole numbers or "simple" fractions (ones with small numbers for numerator and denominator) then we can compute log_{b}(n) and see if the fractional power of p is near to such a fraction.Find out more about logs and how to use them on the Formula for Fibonacci Numbers page

To find out which proper fractions are the nearest to a given decimal number, we need Continued Fractions.

Find out about Continued Fractions on this introductory page :

which is at secondary school level.

So 2021 is not quite as interesting as 2016 but if you can find any other interesting number facts about 2021,

© 2021 Created: 31 December 2020 Dr Ron Knott Home page