@venda said
Do we get to know how many possible solutions there are?.
Bigdogg's logic is sound but I worked it out less logically.
L has got to be 1 because the total cannot be greater than 1,000
S therefor cannot be 1 but could be 2,so I tried to make the sum fit by trying the lowest possible total i.e 1022 and so on
3 solutions in total. I think you just needed to check the possibilities of digits E & S.
This is how I solved it. Perhaps not very elegant, but I think its thorough.
1) T + E = 10*x+S.
Since all digits are 0 -9, "x" may take on two values. namely 0,1 such that the sum ( T+E ) is not greater than 18.
Start by checking the proposition x = 0
2) T + E = 10*0 + S = S
3) E + H = 10*z + S
by 2):
E = S - T
Substitute into 3):
T - H = 10*z
Again the placeholder "z" could be 0 or 1. If we check z = 0 then:
T = H , which is a contradiction as T and H must be distinct.
Check "z = 1"
H - T = 10 , again contradiction, max difference between digits is 9.
So x = 0 is not possible, thus "x" must equal 1 which leads to.
4) T+E = 10*x+S = 10*1 + S = 10+ S
5) E + H + x = E + H + 1 = 10*w + S
Now we can check the possibilities for "w" in a similar manner. Solve 4) for "E". Sub into 5).
10*(w-1) = H - T+1
w= 0, contradiction the least the RHS could be is -8
w = 1 , possible solution:
6) T = H+1
So now to the hundreds place ( w= 1):
L + T + w = L + T + 1 = 10*L + A
Simplify:
7) T + 1 = 9*L + A
We know L <> 0 because its a leading digit, thus L = 1
8) T + 1 = 9 + A
9) T = 8 + A
By substitution of 8) into 9):
10) H = 7 + A
From 9) we see "A" has two possible values. A = 0, 1 because 8 ≤ T ≤ 9 .
However, A = 1 contradicts L = 1 in eq. 7). So we are left with A = 0
We now have ( using what is above)
A = 0
T = 8
H =7
L =1
The two remaining values to find are "E" and "S"
We know from 5) that:
E + H + 1 = 10 + S
Thus ( H =7)
E + 8 = 10 + S
E = S + 2
From here all the possible solutions come from case checking values for E and S against the distinct digit constraint.
{S,E} = { (0,2) , (1,3) , (2,4) , (3,5) , (4,6) , (5,7) , (6,8) , (7,9) }
A = 0, ( 1,2 ) not valid
L = 1, ( 1,3 ) not valid
H = 7, ( 5,7 ) & ( 7,9) not valid
T = 8 , ( 6,8 ) not valid
All solutions ( 3 in total ) are thus:
A = 0
L = 1
H = 7
T = 8
{S,E} = { (2,4) , (3,5) , (4,6) }