 January 2013

# Solution

Well, this is how I worked it out. It took me hours. I am sure you have a shorter method and must have worked it out in minutes. I presume you are here only to cross-check your answer.

First of all, since the sum of two four-digit numbers is a five-digit number, one thing is certain - "T" is a carry over and it can be 1 and nothing else (the carry cannot be greater than or equal to 2). So, T = 1.

Now consider the addition of the left-most digits, either the addition of "E" and "N" will have a carry or not. Suppose there is no carry, then E + N = D, which means R + I = I or R + I = 10 + I (that is the addition has a carry over). If R + I = I, then R = 0 which is not possible. In case R + I = 10 + I, then R = 10 which again is not possible because "R" is a single-digit number. Therefore, one thing is sure - the addition of "E" and "N" has a carry. Therefore, E + N = 10 + D.

This means that 1 + R + I = I (the addition does not have a carry) or 1 + R + I = 10 + I (the addition has a carry). In the first case, 1 + R + I = I. Then, 1 + R = 0. Therefore, R = -1 which is not possible. In the second case, 1 + R + I = 10 + I. Therefore, 1 + R = 10 and R = 9.

Since there is a carry from R + I, we have 1 + A + O = P or 1 + A + O = 10 + P (this addition has a carry). Now A = 6 (given). Therefore, 7 + O = P or 7 + O = 10 + P. Now if 1 + A + O = 10 + P (which means there is a carry), then 1 + R + C = E or 1 + R + C = 10 + E. We know for sure there is a carry, else we will not have "T". So 1 + R + C = E is false and 1 + R + C = 10 + E is true. But R = 9. Therefore, 10 + C = 10 + E which means C = E. This is not possible. Therefore, 1 + A + O = 10 + P is false and 1 + A + O = P is true. That is 7 + O = P (Because A = 6).

So we have established that 7 + O = P. Therefore, P - O = 7. The (P,O) pairs which satisfy this are: (7,0), (8,1) and (9,2) - Since we are dealing with single digits here, pairs like (10, 3), (11,4), etc are not the right choices.. Consider the pair (8,1). We have already established that T = 1. There cannot be two alphabets with the same value. So (8,1) can be ruled out. Likewise R = 9, so (9,2) can also be ruled out. That leaves us with only (7,0). Therefore P = 7 and O = 0.

Phew!

We have found that A = 6, R = 9, O = 0, T = 1 and P = 7. So we are left with the numbers (2, 3, 4, 5, 8). We have seen that if 1 + R + C = 10 + E, then C = E which is not possible. So, R + C = 10 + E. Therefore, 9 + C = 10 + E. Therefore, C - E = 1. The (C,E) pairs from the remaining numbers which satisfy this are (5,4), (4,3) and (3,2). Again, E + N = 10 + D. Let us replace E by 4, 3 and 2 and see what happens. When E = 4, we get N - D = 6. Only (8,2) satisfies this. When E = 3, N - D = 7. There is no pair which satisfies this. Finally, when E = 2, N - D = 8. Again there is no pair which satisfies this. Therefore, (8,2) for the (N,D) pair when E = 4 is true. Thus, E = 4, N = 8 and D = 2.

Phew!

Finally, R + C = 10 + E. Therefore, 9 + C = 10 + 4. Therefore C = 5 and the only remaining number is 3 and the only remaining alphabet is I. Therefore, I = 3.

Oh my god, I have pulled all the hairs from my head. What we finally have is: 9694 + 5038 = 14732.

Some useful links for

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in
• Indian School of Mines, Dhanbad - www.ismdhanbad.ac.in
• Birla Institute of Technology, Ranchi - www.bitmesra.ac.in
• Central Institute of Fisheries Nautical and Engineering Training - www.cifnet.nic.in
• Indian Institute of Information Technology, Allahabad (Deemed University) - www.iiita.ac.in
• Central Marine Fisheries Research Institute, Kochi - www.cmfri.com
• Tata Institute of Social Sciences, Mumbai - www.tiss.edu

DISCLAIMER: The contents of this website are for informational purposes only. Although due care is taken, the correctness and accuracy cannot be guaranteed. The science and medical research articles are for informational purposes only, aimed at kindling an interest in science. They are not intended to provide any kind of advice.