Choose two digits and arrange them to make two double-digit
numbers. Now add your double-digit numbers. Now add your single
digit numbers. Divide your double-digit answer by your single-digit
answer. Try lots of examples. What happens? Can you explain it?
Can you create a Latin Square from multiples of a six digit number?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
There were a number of correct solutions to this problem which
either involved trial and improvement or the solution of a
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
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
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
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
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 60² 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: