Although the Pacific Ocean, Baltic Sea, and Arctic Sea represent three bodies of water, when added together, their sum yields seven Cs. Using the nine digits {0, 2, 3, 4, 5, 6, 7, 8, 9} substitute a different digit for each letter to make a true sum for PACIFIC + BALTIC + ARCTIC = CCCCCCC. Notice that 1 is not a possible digit for this problem. Your task is to provide the value of the word PACIFIC.
Source: mathcontest.olemiss.edu 10/1/2007
SOLUTION
This problem is similar to the problem titled “Peace” dated 3/24/2008.
Step 1
There are two possible choices: . We drop the first choice because the sum of nonzero whole numbers is not equal to zero. Thus, and the carry digit is 1. Remember 1 is allowed as a carry digit.
Step 2
There is only one choice: . Thus, and the carry digit is 2.
Step 3
In this step we cannot use {1, 5, 8} because 1 is not allowed and {5, 8} are already taken.
3.1
All the sums are even, so we move on to the next value for .
3.2
Thus, there is one possible choice, .
3.3
All the sums are even, so we move on.
3.4
All the sums are even, so we move on.
3.5
Remember we cannot use {1, 5, 8}.
Thus, there is one possible choice, .
3.6
Remember we cannot use {1, 5, 8}.
Thus, there are two possible choices,
3.7 Summary
In summary, there are 4 possible choices for :
3.7.1
Carry 
1 

0 


2 


1 





5 
C 
8 
I 
3 
F 
8 
I 
5 
C 

? 
A 
2 
L 
0 
T 
8 
I 
5 
C 

? 
R 
5 
C 

0 
T 

8 
I 

5 
C 

5 
5 
5 
5 
If we choose , then .
Now, we must find what is Carry ? Remember now we cannot use {0, 1, 2, 3, 5, 8}. If you work through all the possible values of , then you will come to the conclusion that this choice has no solution.
3.7.2
Carry 
1 

1 


2 


1 





5 
C 
8 
I 
7 
F 
8 
I 
5 
C 

? 
A 
1 
L 
3 
T 
8 
I 
5 
C 

? 
R 
5 
C 

3 
T 

8 
I 

5 
C 

5 
5 
5 
5 
This choice will force which is not allowed.
3.7.3
Carry 
1 

1 


2 


1 





5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

? 
A 
1 
L 
2 
T 
8 
I 
5 
C 

? 
R 
5 
C 

2 
T 

8 
I 

5 
C 

5 
5 
5 
5 
This choice will also force which is not allowed.
3.7.4
Carry 
1 

2 


2 


1 





5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

? 
A 
? 
L 
7 
T 
8 
I 
5 
C 

? 
R 
5 
C 

7 
T 

8 
I 

5 
C 

5 
5 
5 
5 
Find Carry . Remember now we cannot use {1, 5, 7, 8, 9}.
If you work through all the values of , you will come to the conclusion that there is only one possible choice, .
Step 4 Carry
In this step we cannot use {0, 1, 5, 7, 8, 9}.
Carry 
1 

2 


2 


1 





5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

? 
A 
0 
L 
7 
T 
8 
I 
5 
C 

? 
R 
5 
C 

7 
T 

8 
I 

5 
C 

5 
5 
5 
5 
4.1
If you work through all the possible values of , then you will come to the conclusion that this choice has no solution.
4.2
There is one possible choice, .
4.3
If you work through all the possible values of , then you will come to the conclusion that this choice has no solution.
4.4
There is one possible choice, .
4.5 Summary
In summary, there are 2 possible choices for :
4.5.1
Carry 
1 


1 

2 


2 


1 





3 
A 
5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

? 
B 
3 
A 
0 
L 
7 
T 
8 
I 
5 
C 

3 
A 

6 
R 
5 
C 

7 
T 

8 
I 

5 
C 

5 
5 
5 
5 
5 
This choice will lead to . But, is not allowed because 8 is already taken.
4.5.2
Carry 
1 


1 

2 


2 


1 





6 
A 
5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

? 
B 
6 
A 
0 
L 
7 
T 
8 
I 
5 
C 

6 
A 

3 
R 
5 
C 

7 
T 

8 
I 

5 
C 

5 
5 
5 
5 
5 
This choice gives . Thus, .
Step 5
The only value left is 4, so .
Check
Carry 
1 


1 


1 

2 


2 


1 





4 
P 
6 
A 
5 
C 
8 
I 
9 
F 
8 
I 
5 
C 

2 
B 
6 
A 
0 
L 
7 
T 
8 
I 
5 
C 




6 
A 

3 
R 
5 
C 

7 
T 

8 
I 

5 
C 

5 
5 
5 
5 
5 
5 
5 
Answer: .