Solve Puzzle: Find (p,q,r) Triplets with TELL and DEAL

[K Sengupta]

p, q and r are three positive integers satisfying p< q< r, such that q is the arithmetic mean of p and r.

Determine all possible triplet(s) (p, q, r) that satisfy both the cryptarithmetic equations, where each of the capital letters denotes a different decimal digit from 0 to 9 and, none of T and D can be zero.

r⁴ – q⁴ = TELL, and:
q⁴ – p⁴ = DEAL.

 

[davee123]

I'm not sure if it qualifies as brute force or not-- there were only 19 different things I had to try:

Spoiler

I knew that 5 <= p < 15, so I started assuming that p=5, and worked up from there. Highest you can get with a 4 digit difference was p=12, which turned out to have the only answer:

12,13,14, TELL = 9855, DEAL = 7825

DaveE

 

[Architeuthis Dux]

I would approach this problem by first writing out all the possible combinations of the letters T, E, L, and D and assigning them to different numbers from 0 to 9. For example, T could be 1, E could be 2, L could be 3, and D could be 4. This would give us the equations:

r4 – q4 = 1234, and
q4 – p4 = 1243.

Next, I would use algebraic manipulation to solve for q in terms of p and r. This would give us the equation: q = (p+r)/2. Since q must be a positive integer, this means that both p and r must be either both even or both odd.

Using this information, we can start by plugging in different values for p and r that satisfy the given conditions. For example, if we let p = 2 and r = 4, then q = 3, which satisfies the condition that q is the arithmetic mean of p and r. This would give us the triplet (2, 3, 4) as a possible solution.

We can continue this process by trying different values for p and r, keeping in mind that they must be either both even or both odd. For instance, if we let p = 3 and r = 5, then q = 4, which gives us the triplet (3, 4, 5) as another possible solution.

Using this method, we can generate a list of all possible triplets that satisfy the given conditions. However, it is important to note that there may be other solutions that do not fit this pattern, so it would be necessary to thoroughly check all possible combinations to ensure that all solutions are found.

In conclusion, as a scientist, I would approach this puzzle by first breaking it down into smaller, solvable equations and then using mathematical principles to generate a list of all possible solutions. This approach would ensure that all potential solutions are considered and the puzzle is solved accurately.