Four is Magic is a popular math game. You spell out any number into English:

i.e. $117$ = one hundred seventeen

And then you add the total letters together. So 'one hundred seventeen' has $19$ letters.

When we continue this:

i.e. $117 → 19 → 8 → 5 → 4 → 4$

It is notable that "four" has exactly $4$ letters in it. It is the only number with this quality.

- We exclude counting the spaces and hyphens in the name. Only letters.
- We don't say "one hundred
one". We exclude the "and".*and*

I came up with a bonus feature to this puzzle.

Instead of adding the words count of each word like this:

i.e. $117$ = one hundred seventeen = $3 + 7 + 9 → 19$

We will be taking the product of each word count:

i.e. $117$ = one hundred seventeen = $3 * 7 * 9 → 189$

This leads to very different results! $4$ is still special, but now are there other numbers with this quality?

Yes, there are!

- $24 →$ twenty four $= 6 * 4 = 24$

Can you find the others?

- Problem 1: I have found 9 solutions for $a → a$
- Problem 2: I have found 25 other solutions for $a → b → ... → a$

If you can find solutions for $a → a$, be sure to post them in this sequence: A058230.

How can you code this problem?

What are the solutions to both problems?
My 9th solution is a monstrous 42 digit number!

Are there more than 9 solutions?

What other interesting numbers can you find?

Turns out these numbers are called Fortuitous Numbers.

No vote yet

1 vote

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestFascinating! I just might try coding this. I'll probably even use this digit-separating formula I came up with a while back.

Log in to reply

Thank you! It's a really interesting programming challenge. That's also a very useful formula!

Log in to reply

Phew! 100 lines of code later, and it takes me a minute to reach the fourth OEIS entry. :)

Log in to reply

Log in to reply

Does that mean that you found more entries than are listed in the OEIS sequence, or am I reading that wrong?

Log in to reply

Yes, I found 3 more, the largest of which has 42 digits. I’m trying to add the new terms to OEIS, but now I am waiting for approval from an editor.

I’m also adding a new sequence focused on numbers of type “a -> b -> ... -> a”. But that one also needs to be approved by an editor.

Log in to reply

Woah, cool! How long did that take?

Log in to reply

But then no new solutions appeared after 24 hours of running, up to about ~ 10^130

Log in to reply

Log in to reply

Log in to reply