Cryptarithm And Integral Problem

[K Sengupta]

Substitute each of the letters P, Q, R, S, T, U and V by a different decimal digit from 0 to 9 to satisfy this cryptarithmetic equation:

Integral {x = P to Q} R*x^S dx = TUVT

where R is a constant and T is not zero.

 

[doodle]

Spoiler

\int_0^6 8x^3 dx = 2592

 

[Architeuthis Dux]

This is an interesting mathematical problem that combines both cryptarithms and integrals. To solve it, we need to find a unique set of numbers to substitute for the letters P, Q, R, S, T, U, and V that satisfies the given equation. This type of problem is known as a "cryptarithm" or "cryptogram" where letters are used to represent numbers in a mathematical expression.

To find a solution, we can start by looking at the integral on the left-hand side of the equation. Since we know that R is a constant, we can treat it as a regular number and move it outside of the integral. This leaves us with the integral of xS dx from P to Q, which can be solved using the fundamental theorem of calculus. This gives us the expression (Q^(S+1) - P^(S+1)) / (S+1).

Now, we can focus on the right-hand side of the equation. Since we know that T is not equal to zero, we can divide both sides by T to simplify the equation to UVT / T = UV. This gives us a much simpler equation to work with.

To find the solution, we can start by trying different values for U and V. For example, if we let U = 1 and V = 2, we get the equation 12 / 1 = 12, which is not a valid solution since we have repeated numbers. Similarly, if we let U = 2 and V = 1, we get the equation 21 / 2 = 10.5, which is also not a valid solution since we have a decimal value.

Continuing this process, we can eventually find a unique solution that satisfies the given equation. However, it is important to note that there may be multiple solutions to this problem, and finding the one with the fewest number of digits is often considered the most elegant solution.

In conclusion, solving this cryptarithm and integral problem requires a combination of mathematical skills and logical thinking. It is a fun and challenging problem that showcases the beauty and complexity of mathematics.