Tis unique
Problem
| * | * | 4 | |
| + | 2 | 8 | * |
| * | * | * | * |
This addition sum uses all ten digits 0, 1, 2 ...9 exactly once. Find the sum and show that the one you give is the only possibility.
Student Solutions
Congratulations to all the following for their correct solutions:
Andrei Lazanu Lazanu (School number 205, Bucharest),
Fiona Watson (Stamford High School),
Musab Khawaja,
Chong Ching Tong, Chen Wei Jian and Teo Seow Tian (River Valley
High School, Singapore),
Clement Goh Tian (River Valley High School, Singapore),
Emma Holmes
It is good to see you explain how you came to your answer. Some of you said you solved the problem by "trial and error" but I am sure you made some decisions along the way. Sometimes it is necessary to try several possible numbers before finding the one that works for other constraints. Next time try to explain your thinking. I have included Andrei's solution as one of those that gave a very complete explanation of how they thought the problem through.
First I observed that the first digit of the result must be 1:
| * | * | 4 | |
| + | 2 | 8 | * |
| _ | _ | _ | _ |
| 1 | * | * | * |
I used for the other letters the following notation:
| a | b | 4 | |
| + | 2 | 8 | c |
| _ | _ | _ | _ |
| 1 | d | e | f |
2 + a + n = 
a + n = 8 + d, where a and d are smaller than 10.
| 1.1. | n = 0 |
a = 8 + d.
a can be 8 or 9 and d 0 and 1 respectively, but in the first case 8
was used, and in the second case 0 was used.
| 1.2. | n = 1 |
a + 1 = 8 + d
a can be 7, 8 and 9 and d can be 0, 1 and 2. In the first situation
it works because the digits weren't used another time, in the
second combination it doesn't work, and the last situation doesn't
work. The result is:
| 7 | b | 4 | |
| + | 2 | 8 | c |
| _ | _ | _ | _ |
| 1 | 0 | e | f |
4 + c = 
c can be : 3, 5, 6, 9. Then
c = 5; f = 9
c = 9; f = 3
| 2.1. | c = 5; f = 9 |
| 7 | b | 4 | |
| + | 2 | 8 | 5 |
| _ | _ | _ | _ |
| 1 | 0 | e | 9 |
b + 8 = 
Using only 3 and 6 there isn't any possibility.
| 2.2. | c = 9; f = 3 |
| 7 | b | 4 | |
| + | 2 | 8 | 9 |
| _ | _ | _ | _ |
| 1 | 0 | e | 3 |
b + 8 + 1 =
b + 9 =
In this situation only b = 6 and e = 5 satisfies the condition.
This is the only solution for the problem, because using a step-by-step method, I obtained only one solution.
The addition is:
| 7 | 6 | 4 | |
| + | 2 | 8 | 9 |
| _ | _ | _ | _ |
| 1 | 0 | 5 | 3 |