# What Is a Dudeney Number?

*I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.*

## What Are Dudeney Numbers?

A Dudeney number (named after the English mathematician, Henry Dudeney, who used them in puzzles he created) is a positive integer such that the sum of its separate digits is equal to its own cube root.

For example 512 is a Dudeney number, because if we add its digits together we get 5 + 1 + 2 = 8 and 512 = 8^{3}.

## How Many Dudeney Numbers Are There?

Including the trivial case of 1, there are only six Dudeney numbers. These are 1, 512, 4913, 5832, 17 576 and 19 683. We can check this by summing their digits.

1^{3} = 1

(5 + 1 + 2)^{3} = 8^{3} = 512

(4 + 9 + 1 + 3)^{3} = 17^{3} = 4913

(5 + 8 + 3 + 2)^{3} = 18^{3} = 5832

(1 + 7 + 5 + 7 + 6)^{3} = 26^{3} = 17 576

(1 + 9 + 6 + 8 + 3)^{3} = 27^{3} = 19 683

## Who Was Henry Dudeney?

Henry Dudeney (1857-1930) was an English mathematician and author, best known for his publications on mathematical puzzles and problems.

Although he had only a basic education and no formal mathematics training, he was prolific in writing articles for magazines featuring mathematical and chess problems. One of his best know problems is the 'Haberdasher's Problem' which looks at how an equilateral triangle can be chopped into four pieces in such a way that can be reassembled into a square. You can see an image of the solution below.

The Dudeney numbers are named in his honour as he discussed them in one of his puzzles named 'Root Extraction'.

## Converting a Square Into an Equilateral Triangle

## Social and Amicable Dudeney Numbers

We call a number a Social Dudeney number if, when we sum its digits and cube this sum, we then get another number where we can repeat this process and keep going in a chain until we get back to our original number. If there are only two Social Dudeney numbers in the chain, we call them Amicable Dudeney Numbers.

For example take the number 6859.

(6 + 8 + 5 + 9)^{3} = 28^{3} = 21 952

Repeat the process with 21 952.

(2 + 1 + 9 + 5 + 2)^{3} = 19^{3} = 6859

Adding the digits and cubing the sum keeps us bouncing back and forth between 6859 and 21 952, hence these are Amicable Dudeney numbers (note that they themselves are not Dudeney numbers under the normal definition).

## Generalising Dudeney Numbers

There are two ways in which we can generalise the process of finding Dudeney numbers. So far we have been looking at cubes in the base ten decimal number system, but what about if we changed powers or base?

For example, remaining with base 10, but using a power of 4 we can find six Dudeney numbers: 1, 2401, 234 256, 390 625, 614 656 and 1 679 616.

Let's check this for 234 256.

(2 + 3 + 4 + 2 + 5 + 6)^{4} = 22^{4} = 234 256 so using a power of 4 we get back to where we started. Try checking the others for yourself.

We can also create Dudeney numbers in other bases. For example, in base 8 and using a power of 3, there are three Dudeney numbers: 330, 4225 and 5270. See if you can work out how to check these.

## A Similar Number: The Kaprekar Number

A Kaprekar number (named after the Indian maths teacher D. R. Kaprekar, 1905 - 1986) is another type of number with an interesting link between its original form, its digits and powers.

For any given base, we call a number a p-Kaprekar number if we can split the digits of the square of the number up into two parts, with the second part having p digits, and then add the two parts together to get back to the original number.

For example 55 is a 2-Kaprekar number because 55^{2} = 3025 and 30 + 25 = 55.^{}

The smallest Kaprekar numbers in base 10 are 1 (1^{2} = 1 and 0 + 1 =1) and 9 (9^{2} = 81 and 8 + 1 = 9) and there are infinitely many Kaprekar numbers larger than this.

## Sources and Further Reading

- Aizu Online Judge - Dudeney Number
- Geeks for Geeks - Dudeney Numbers
- Jakob.at - Generalized Dudeney Numbers
- Wikipedia - Dudeney Number
- Wikipedia - Henry Dudeney
- Maths History St Andrews - Henry Dudeney
- IITB - Kaprekar Numbers

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 David**