Substitute digits in place of letters

1,260.0K Views

Substitute digits in place of letters to make the following relation true.

S T I L L
+ W I T H I N
——————–
L I M I T S

Note
The leftmost letter cant be zero in any word.
There must be a one-to-one mapping between digits and letters. e.g. if you substitute 3 for the letter S, no other letter can be 3 and all other S in the puzzle must be 3.

SherlockHolmes Expert Asked on 19th October 2015 in

So obviously this can be solved by brute force–trying different values for each letter.  The challenge is how to deduce the answer with as little trial and error as possible.  We can narrow it down with some observations.

Both I’s in LIMITS line up under an I, so let’s start there.
Because S can’t be zero and S + I results in I as the last digit, we know T + T >= 9 (to generate carry) and that S = 9.
Since S = 9, that must mean that H = 0 so that H + I = I with no carry.
Since S = 9, L + N = 9.
There is no carry from L + N, so L + I = T or 10 + T, but we know there is also no carry from L + I and that I is not zero.
Because T + T = 10 + M and M can’t be zero, then T isn’t 5.
Because WITHIN and LIMITS are both six digits, we know that L = W + 1.

So before any trial and error we have:

S = 9
H = 0
T = {6,7,8} = L + I
M = {2,4,6} = 2*T – 10
W = {1,2,3,4,5,6} = L – 1
L = {2,3,4,5,6,7} = W + 1 = 9 – N
N = {2,3,4,5,6,7} = 9 – L
I = {1,2,3,4,5,6} = T – L

Looking at this, we see that L has a relationship to every other letter, so let’s look at that.  Since W = L – 1 and N = 9 – L, L can’t be 5 since that would require both W and N to be 4.  So now the possibilities are:

S = 9
H = 0
T = {6,7,8} = L + I
M = {2,4,6} = 2*T – 10
W = {1,2,3,5,6} = L – 1
L = {2,3,4,6,7} = W + 1 = 9 – N
N = {2,3,5,6,7} = 9 – L
I = {1,2,3,4,5,6} = T – L

Now we’re ready for trial and error.  Since there is only one way to assign 8 and since eight out of ten digits are used (only two unused), try assigning T = 8, which gives M = 6.  Since 6 is taken, neither L nor I can be 2 (since 8 – 6 = 2) nor 4 (since they need to be different and sum to 8).  Also, since L = 9 – N and N can’t be 6, then L can’t be 3 and L = W + 1 and W can’t be 6, so L can’t be 7, leaving no possible value for L.  So we know T can’t be 8, leaving:

S = 9
H = 0
T = {6,7} = L + I
M = {2,4} = 2*T – 10
W = {1,2,3,5} = L – 1
L = {2,3,4,6} = W + 1 = 9 – N
N = {3,5,6,7} = 9 – L
I = {1,2,3,4,5} = T – L

There are only two ways to assign 7, so try assigning T = 7, which gives M = 4.  Since 4 is taken, neither L nor I can be 3 or 4.  Since N can’t be 7 and N = 9 – L, L also can’t be 2, leaving only L = 6:

S = 9
H = 0
T = 7
M = 4
W = 5
L = 6
N =3
I = 1

This give a unique assignment for each letter, so see if it works:  97166 + 517013 = 614179
It does work, on only the second (of three possible) guesses.

dougbell Genius Answered on 21st October 2015.

• More puzzles to try-

• A Man and his wife

A man who wanted to prove his love to his wife. So, he climbed the highest mountain, swam the deepest ...Read More »
• Mouth is so large

On my own, I am darkness, a black abyss. But, Life brings me light with its gentle kiss. I am ...Read More »
• Divide barrels and wine evenly

A man is the owner of a winery who recently passed away. In his will, he left 21 barrels (seven ...Read More »
• Should you stay, or switch, or doesn’t it matter?

The host of a game show, offers the guest a choice of three doors. Behind one is a expensive car, ...Read More »
• Who visited Which place puzzle

In Mulund, the shoe store is closed every Monday, the boutique is closed every Tuesday, the grocery store is closed ...Read More »
• How fast to drive to catch the METRO train ?

Mr Albert go to his office by metro. However nearby metro station is quite far from his place and he ...Read More »
• Pick a number!!

What is 2+2? What is 4+4? What is 8+8? What is 16+16? Pick a number between 12 and 5.Read More »
• It has no heart but yet it lives riddle

It has no heart but yet it lives. If it does not live then you won’t live. It does not ...Read More »
• How could only 3 apples left

Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »
• What is the current Temperature of London

This Rebus represents the current Temperature of London 0 ____________ B.Tech M.B.B.S What is the current Temperature of London?  Read More »
• Others Memories

It has memories, but none of its own, whatever is on my inside is what is shown. If it is ...Read More »
• Weightless Elephant

What is bigger than an elephant yet weighs nothing. What is it?Read More »
• Geographical family

A cloud was my mother, the wind is my father, my son is the cool stream, and my daughter is ...Read More »
• Goes through a door

It goes through a door, but never goes in or comes out. What is it?Read More »
• How did the man died?

In the locked room there were no widows, nothing and a man is hung from the ceiling and a puddle ...Read More »
• Surgeons\’ favorite game

What is a surgeons\’ favorite game?Read More »
• What shape completes the bottom line?

triangle pentagon square square hexagon hexagon square pentagon hexagon hexagon hexagon square triangle hexagon octagon octagon octagon octagon ==?==Read More »
• The Man You Hold Dear

There’s a day, that comes around once a year. To celebrate the man, you hold dear. He gave you life, ...Read More »
• Compose the fraction riddle

Use the digits below once each only to compose two fractions which when added together equal 1. 0,1,2,3,4,5,6,7,8,9Read More »
• Determining a hidden digit

Think of a positive integer, call it X.  Shuffle the decimal digits of X, call the resulting number Y.  Subtract ...Read More »