For the last couple of days, I've been asking you to create a new puzzle. Today, I'd like you to describe HOW to create a puzzle with a unique solution, or two solutions, or three. How did Don Steward create these and KNOW that they only had one solution?

# A puzzle is defined:

ReplyDeleteA "solution"

A, B

C, D

exists such that

W = A*B

X = C*D

Y = A*C

Z = B*D

where `W`, `X`, `Y`, `Z` are given.

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Let's define the "Solution Product" as the product of two rows or two columns.

SP = W*X = Y*Z = A*B*C*D

The Solution Product is constant,

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# Suppose there exist two unique solutions to the puzzle, `n` and `m`

Soluton `n`:

An, Bn

Cn, Dn

Solution `m`:

Am, Bm

Cm, Dm

# Case 1 (Theorem: `An != Am`)

Suppose `An = Am`

Since `W = An*Bn = Am*Bm`, it follows that `Bn = Bm`, and so `Cn = Cm` and `Dn = Dm`, and so the solutions are NOT unique. (This would imply later a common factor of `1`).

Therefore, `An != Am`, and so either `An < Am` or `An > Am`.

# Case 2 (& 3)

Suppose `An < Am` (If reversed, follows similarly.)

Since `W = An*Bn = Am*Bm`, there must be some number `Fnm > 1` such that `Am = An*Fnm`. It follows that Bn = Bm*Fnm, and therefore `Bm < Bn`.

Similarly, this `Fnm` factor passes between the `C` and `D` elements of the `n` & `m` solutions.

The two solutions can be written in relation to each other and this common factor thusly:

Soluton `n`:

An, Bm*Fnm

Cm*Fnm, Dn

Soluton `m`:

An*Fnm, Bm

Cm, Dn*Fnm

The Solution Product written in these terms is

SP = An*Bm*Cm*Dn*Fnm^2

# Conclusion

For a prime factorization of `SP`, there will exist at least 1 prime with exponent 2 or greater. The product of these primes are potential values of `Fnm`.

In case 1, `Fnm = 1`, therefore the two solutions were not unique. If a puzzle is defined to have the property that it has only one solution, then no common factor exists for either of the solution's diagonal pairs since the only allowable `Fnm` can be `1`.