Double with 1 to 9
Can you find a number and its double using the digits $1$ to $9$ only once each?
Age
11 to 14
Challenge level
Problem
Using each of the numbers 1 to 9 once and once only, find two whole numbers, one of which is double the other.
Can you find more than one solution?
Student Solutions
Answer: There are 12 possible pairs:
6729 & 13458
6792 & 13584
6927 & 13854
7269 & 14538
7293 & 14586
7329 & 14658
7692 & 15384
7923 & 15846
7932 & 15864
9267 & 18534
9273 & 18546
9327 & 18654
There are 9 digits altogether so one number will have 4 digits and one will have 5 digits.
The larger number will start with a 1 and the smaller number will start with a number at least 5
The larger number will be even
Try:
6
1 2 3, 4, 5, 7, 8, 9 left
either or
6 4 6 7
1 2 8 3, 5, 7, 9 left 1 2 4 3, 5, 8, 9 left
Stuck! Only odd numbers left but Stuck! 5 doubles to 0 or 1 so cannot
1 has not been carried from ones go on top. But need 7 or 2 on top for
digits! 5 on bottom. Both have been used!
Start again. Try:
6
1 3 2, 4, 5, 7, 8, 9 left
how about:
6
1 3 5 2, 4, 7, 8, 9 left
6 7
1 3 5 2, 4, 8, 9 left
either or
6 7 8 6 7 9
1 3 5 2, 4, 9 left 1 3 5 8 2, 4 left
Stuck! 8 doubles to give 6 or 7 Perfect! 6792 and 13584
There are lots of possibilities.
The table below shows all of the possible starting digits for the four digit and five digit numbers.
| smaller number | larger number |
|---|---|
| 6 _ _ _ | 1 3 _ _ _ |
| 7 _ _ _ | 1 4 _ _ _ |
| 7 _ _ _ | 1 5 _ _ _ |
| 8 _ _ _ | 1 6 _ _ _ |
| 8 _ _ _ | 1 7 _ _ _ |
| 9 _ _ _ | 1 8 _ _ _ |
Working systematically through each row in the table and finding all of the possible pairs for that row, these are the 12 possible pairs:
6729 & 13458
6792 & 13584
6927 & 13854
7269 & 14538
7293 & 14586
7329 & 14658
7692 & 15384
7923 & 15846
7932 & 15864
9267 & 18534
9273 & 18546
9327 & 18654