Math Riddle for those who are extra clever?

Question

okay, it's sort of a guess and check..

here's the problem:
STOP
- TOPS
------------
POTS

okay, so the letters each mean a single digit number, and the problem is subtraction. you can do borrowing, etc, but NO negatives allowed! you can use zeros. (basically numbers zero through nine)

PLEASE I NEED THIS BY THURSDAY!!!

[this is in case the problem looks weird: it's STOP minus TOPS equals POTS]

P.S. I also posted this in the polls and surveys section, so that's how desperate i am, since i wasted 20 points. I NEED THE ANSWER!!!

okay so one letter is equal to ONLY ONE NUMBER! lol sorry I should have mentioned that before...

Answer 1

Are the digits all different? I'll assume they are for now.

It's easier to rephrase it as an addition problem:
POTS + TOPS = STOP

You know that S + S ends in P by the last digit. Then P must be an even digit: 0,2,4,6,8. From the first digit, we see that S>=P+T. In particular, then S>P. So the pair (S,P) can only be one of: (5,0), (6,2),(7,4),(8,6) or (9,8).

Now, if (S,P)=(5,0), then we see that:

TO05 + 0OT5 = 5TO0. But then since T can't be 5, T must be 4. And then O must be 0+T+1 = 5, which contradicts are assumption that the digits are different.

If (S,P) = (6,2):

TO26 + 2OT6 = 6TO2. From the first digit, we see that T must be 3 or 4. In particular, then in the tens place, 2+T+1 doesn't cause a carry, so in the hundreds place, O+O ends in T. That means T must be even, so T=4, O=2 (since O+O can't carry if T=4.) But then O+P.

(S,P) = (7,4):

TO47 + 4OT7 = 7TO4

T must be 2 or 3 for the 1000s digit to come out right.
But then in then tens digit, T+4+1<10, so O+O ends in T.
So T must be even, T=2, and O=6 (since O+O must carry.)
That works out to:

2647 + 4627 = 7274, not 7264.

(S,P)=(8,4):

TO68 + 6OT8 = 8TO6

T must be 1 or 2 for the 1000s place to work.

But then in the tens place T+6+1 <10, so again O+O ends in T and T must be 2. Then O=1. But that doesn't work, either, since T+6+1=9 doesn't equal O.

Finally, (S,P) = (9,8):

TO89 + 8OT9 = 9TO8

Now, T=0, 1. If T=0, then T+8+1 = 9 must be O from the tens digit, which is not possible (Since O!=S).

If T=1, then O=0, and we finally have our solution:

1089 + 8019 = 9108

Or, in your original equation:

9108 - 1089 = 8019

Answer 2

9108
- 1089
---------
8019

LOGIC:
Rewrite:
POTS + TOPS = STOP

1) Looking at the first digits
Either P + T = S
or P + T + 1 = S if O >= 5

2) Looking at the 3rd digits:
If S >= 5
Either T+P+1 = O
or T+P+1 = 10+O which gives T+P = 9+O
In this case O = 0 and S = 9
If S = 9, P = 8, T = 1
And we find that this works.

3) I haven't tested the other cases though since I got lucky on my first test.

Answer 3

..POTS
+TOPS
-----------
..STOP

P=even
lS-Ol = 1
either S<5, S=O+1, O<4
or S>=5, S=O-1, O>=6
P+T<10,
P then S>=5, O>=6
5<=S<=9,
6<=O<=9, O=0
2O<10, O<5
O=0, S=9, P=8
8+T<10,
T<2
T =/ 0 (T inequal 0)
T=1