Basically

There were a number of correct solutions to this problem which either involved trial and improvement or the solution of a quadratic equation.

Correct solutions were received from Michael Brooker, Yatir Halvi (Maccabim-Reut High School), Ross Haines, Robert Haynes and Andre Lazanu. Well done!

Ross Haines first explained how number bases work:

We use base ten, which is this:

This can also be written as

The second way is the way you write bases. It is the same format for all of them. For example, base 4 would be written as:

To work a number from a base to base 10 you times the number by the column it is in. eg, using the number 321 in base four

Which gives:

1 x 1 equals 1
2 x 4 equals 8
3 x ( 4 squared) equals 48

so 321 in base 4 would be 1+8+48=57 in base 10

Here is the trial and improvement solution submitted by Michael:

I worked it out like this: I decided to begin with bases much higher than 10, as 123 is much smaller than 3723. By trial and error I discovered that the answer was somewhere between 40 and 80.

So I tried translating "123" in Base 60 into Base 10. This time I did get 3723. Using powers I worked out the value of each base-60 digit in base 10 and added the figures together:

"3", the digit on the right, is the same in decimal

"2" x 60' gives 120

"1" x 60 ² gives 3600

Total = 3723

Yatir and Andrei found the solution by solving the quadratic equation 3x^2+2x+3=3723.

This works because the values of the columns in base x are: