## 02 August 2011

### "Pretty wild narcissistic numbers"

A pretty wild narcissistic number is an integer N that can be represented by mathematically manipulating its digits (in order) using only the arithmetical operations of + - x ÷  ^  √  ! .
Examples shown above, for those who enjoy recreational mathematics.   Found at archery, which has a link to many more examples.

1. I think the use of square roots is cheating, because every square root contains an implicit "2".

2. It looks like they don't use any "implicit" numbers. In the examples, 71 and 936 don't have 2s.

3. That's my point: they leave out the implicit 2s, which to me feels like cheating.

To put it another way, the class of number they've invented is defined in terms of the superficialities of standard notation with all its quirks, as opposed to what the expressions actually mean.

That just feels somewhat inelegant.

4. Maybe Friedman numbers would satisfy you ;) they're the same but with only the basic set of operators allowed.

I agree that using the square root is a bit like cheating, as does the factorial operator since the expression n! "implies" the use of {1,2,...,n-1}

I wonder if there are any binary Friedman numbers? has to be... I'm gonna see if I can find one. Thanks Stan, I was looking for a nerdy way to waste a few minutes ;)

5. outerhoard, I see what you mean.

6. What if we let A ^/ B represent the B root of A (i.e. A to the power of the reciprocal of B).

Perhaps we should also allow logarithms, using A ?^ B to represent the base B logarithm of A (or A is what power of B). Roots and logarithms are both inverse operations of exponentiation, so I'd say there's a case for allowing them both.

For example:
1000 ^/ 3 = 10
1000 ?^ 10 = 3

Factorials I can take or leave. The numbers in an expanded factorial are not implicit in quite the same way as a two is implicit in a square root, because you can't write out n! in full for unknown n. On the other hand, it's rather arbitrary to pick out factorials in particular as an allowable operation.

I wrote a blog post once about what buttons I would include if I designed my own scientific calculator.