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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two 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: