@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) }