An alien base

Number Theory Level pending

A paper drops out of the sky with the following alien equation scribbled on it:

$D + B = BE = (B + B + B + B)!$

You have no idea what base they are working in, and you have no idea what digits each of the letters represents.

However, you do know that by some miracle, all of the symbols have the same meaning as they do here on earth (e.g. "plus", "equals", "times", "factorial", "parenthesis").

What is the value of the alien number $BED$ when written in decimal (and our normal earthling digits $0-9$ )?

If you think there is not enough information, enter the value 99999.

Image credit: http://images.frompo.com/

The answer is 599.

1 solution

Geoff Pilling
Nov 17, 2016

From the left side of the equation you can tell that $B=1$ and $E = 0$ . (In any base, if two single digit numbers are added together to a two digit number, the first digit of the two digit number must be a $1$ , and since $B$ is $1$ , $BE$ must be the first two digit number in that base, or $10_n$ where $n$ is the alien base)

From the right side, you know that $B+B+B+B = 4_{10}$ so $(B+B+B+B)! = 24_{10} = 10_n$ , where $n$ is the alien base.

So, we are working in base $24$ , and so from the left half of the equation, you can see that $D = 23_{10}$ .

So, the value for $BED$ would be:

$24^2+0+23 = \boxed{599}$

Note: This problem was inspired by @Calvin Lin who gave me the idea to write a problem about an alien base where you don't know what digits the symbols represent.

An alien base

The answer is 599.

1 solution

Note: This problem was inspired by @Calvin Lin who gave me the idea to write a problem about an alien base where you don't know what digits the symbols represent.

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