Base what?

Number Theory Level 4

Suzie McTube decided to invent a unique number system. It's not base 2. It's not base 3, or even base 4, 5, 6 or 7... But it's a combination of all the bases!

It works like this: The first digit (farthest on the right, or "unit's digit") is base 2. The second digit (again from the right) is base 3. The third is base 4 etc. So, for example, the first few numbers are:

$1, 10, 11, 20, 21, 100, 101, 110, \ldots$

So, if 21 is the $5^\text{th}$ number (as shown above), what is the $99^\text{th}$ number? (99 is in base 10 or decimal)

Inspired by : Ujjwal Rane .

The answer is 4011.

1 solution

Geoff Pilling
Aug 30, 2016

In general, if the digits (from right to left) are given by $x_1$ , $x_2$ , $x_3$ , $x_4$ , ...

Then, the value of a number is, $\sum{n!x_n}$

$99 = 4*4! + 0*3! + 1*2! + 1*1!$

So the $99th$ number is $\boxed{4011}$

@Geoff Pilling Beautiful problem! And thank you for that kind note about inspiration. These numbers are also very handy in programming. I create logic board games for fun and these numbers make very compact code and efficient algorithms. Used them extensively in my work too.

Ujjwal Rane - 4 years, 9 months ago

Woo hoo... Thanks, Ujjwal... And, I see you solved it! Its always cool when someone finally solves a problem... Until then, I'm never quite sure if I made a mistake in the question... Anyway, nicely done! :0)

Geoff Pilling - 4 years, 9 months ago

Base what?

Inspired by : Ujjwal Rane .

The answer is 4011.

1 solution

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