# Logical expression puzzle

A through I are all binary variables (so they're either true of false).

How much is the binary number 0bABCDEFGH if all of these are true?

• (A or B or C) and (D or E or F) and (G or H or I)
• (C and D and F) or (A and I and G) or (B and E and H)
• (B <=> C) and (A <=> F) and (I <=> B)
• (I xor F) and (C xor D) and (E xor G)
• A => F => I => B

(please also provide feedback how hard it was to solve)

• You may want to clarify your question, as it stands, it doesn't make much sense and it doesn't look like much of a puzzle but rather a math problem. Jan 19, 2017 at 20:40
• I added "if all of these are true"? try to think about it for a bit and you will see what you need to solve. Jan 19, 2017 at 20:44
• Did you deliberately say 0bABCDEFGH rather than 0bABCDEFGHI? Jan 19, 2017 at 21:57
• 0bABCDEFGH is nice 8 bit number, originally this didn't have binary number mentioned in the question, nor explicitley stated that letters are all binary variables si this was kind of hidden hint, or confirmation Jan 20, 2017 at 18:59

It seems

105

is the solution, when

B, C, E, H and I are true, and A, D, F and G are false.

Explanation:

Rule 3: I, B, C have the same value (call it X), and A and F as well (call it Y).
Rule 4: I xor F must be true; this implies X != Y.
Rule 4: C xor D must be true; this implies D = Y.
In rule 2, (C and D and F) can never be true as C = X and D = F = Y.
In rule 2, (A and I and G) can never be true as A = Y and I = X.
Rule 2: So (B and E and H) must be true. This implies B = X = true.
Rule 4: E xor G implies G = false.

Note that

we haven't used the first and last conditions; therefore, they can be removed while keeping the solution.