Directions for questions 62 and 64: Substitute different digits (0, 1, 2, ... 9) for different letters in the problem below, so that the corresponding addition is correct and it results in the maximum possible value of MONEY.

P A Y
M E
R E A L
M O N E Y

This grid represents the addition PAY + ME + REAL = MONEY, where each letter stands for exactly one digit and no two letters share the same digit.

The letter 'Y' should be:

Question

If a cryptanalyst only knows the encryption algorithm and ciphertext, then which type of attack can be performed by him?

Answer 1

This is a cryptarithm, PAY + ME + REAL = MONEY, where each letter is a different digit from 0 to 9, and we want the digit that Y must take in the arrangement that gives the biggest possible value of MONEY.

Since PAY, ME and REAL together can add up to a little over 11000 at most, the sum MONEY, a 5-digit number, can only start with M = 1.

Looking at the units place, Y + E + L must end in Y again, which is only possible if E + L = 10, carrying a 1 into the tens place.

Looking at the tens place, A + M + A plus the carried 1 must produce the digit E, with another carry of 1 going into the hundreds place. With M = 1, this gives $2A + 2 = E + 10$, so $2A = E + 8$.

Looking at the hundreds place, P + E plus the carried 1 gives the digit N.

Trying the different values that satisfy $2A = E + 8$ while keeping every letter a distinct digit, the choice E = 2 is the one that lets every remaining letter be filled in with a different digit and produces the largest possible MONEY. This gives A = 5, since $2A = 10$, and L = 8, since $E + L = 10$. Filling in the rest of the letters so that all nine are distinct gives P = 3, R = 4, O = 9 and N = 6, making MONEY = 19627.

Let's summarize:

The units and tens columns force E + L = 10 and 2A = E + 8.
E = 2 is the value that lets the whole grid be filled with nine distinct digits while keeping MONEY as large as possible.
This gives the completed value MONEY = 19627.

So in this maximum arrangement, the letter Y takes the digit 7.

Answer 2

This is a cryptarithm, PAY + ME + REAL = MONEY, where each letter is a different digit from 0 to 9, and we want the digit that Y must take in the arrangement that gives the biggest possible value of MONEY.

Since PAY, ME and REAL together can add up to a little over 11000 at most, the sum MONEY, a 5-digit number, can only start with M = 1.

Looking at the units place, Y + E + L must end in Y again, which is only possible if E + L = 10, carrying a 1 into the tens place.

Looking at the tens place, A + M + A plus the carried 1 must produce the digit E, with another carry of 1 going into the hundreds place. With M = 1, this gives $2A + 2 = E + 10$, so $2A = E + 8$.

Looking at the hundreds place, P + E plus the carried 1 gives the digit N.

Trying the different values that satisfy $2A = E + 8$ while keeping every letter a distinct digit, the choice E = 2 is the one that lets every remaining letter be filled in with a different digit and produces the largest possible MONEY. This gives A = 5, since $2A = 10$, and L = 8, since $E + L = 10$. Filling in the rest of the letters so that all nine are distinct gives P = 3, R = 4, O = 9 and N = 6, making MONEY = 19627.

Let's summarize:

The units and tens columns force E + L = 10 and 2A = E + 8.
E = 2 is the value that lets the whole grid be filled with nine distinct digits while keeping MONEY as large as possible.
This gives the completed value MONEY = 19627.

So in this maximum arrangement, the letter Y takes the digit 7.

Over Number	Run Rate
N-2	8.00
N	7.43
N+2	8.11
N+4	8.45
N+6	8.08

Show Hint

The Correct Option is D

Solution and Explanation

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