Puzzle for September 24, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
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Hint #1
eq.2 may be written as: F = B + A + C In the above equation, replace A + C with B + D (from eq.4): F = B + B + D which becomes eq.2a) F = 2×B + D
Hint #2
In eq.3, replace F with 2×B + D (from eq.2a): E - D = 2×B + D - E Add D and E to both sides of the above equation: E - D + D + E = 2×B + D - E + D + E which becomes 2×E = 2×B + 2×D Divide both sides by 2: 2×E ÷ 2 = (2×B + 2×D) ÷ 2 which becomes eq.3a) E = B + D
Hint #3
eq.2a may be written as: F = B + B + D In the above equation, replace B + D with E (from eq.3a): eq.2b) F = B + E
Hint #4
In eq.2b, substitute C + E for F (from eq.1): C + E = B + E Subtract E from each side of the equation above: C + E - E = B + E - E which makes C = B
Hint #5
In eq.4, substitute B for C: A + B = B + D Subtract B from each side of the above equation: A + B - B = B + D - B which makes A = D
Hint #6
Substitute B + D for A + C (from eq.4), and 2×B + D for F (from eq.2a) in eq.5: B × E = B + D + D + 2×B + D which becomes B × E = 3×B + 3×D which may be written as eq.5a) B × E = 3×(B + D)
Hint #7
In eq.5a, replace (B + D) with E (from eq.3a): B × E = 3×E Divide both sides of the above equation by E: (B × E) ÷ E = (3×E) ÷ E which makes B = 3 and also makes C = B = 3
Hint #8
In eq.6, replace B and C with 3, and D with A: 3 × 3 = A ÷ A + F which becomes 9 = 1 + F Subtract 1 from both sides of the above equation: 9 - 1 = 1 + F - 1 which makes 8 = F
Hint #9
In eq.2a, substitute 8 for F, and 3 for B: 8 = 2 × 3 + D which becomes 8 = 6 + D Subtract 6 from both sides of the equation above: 8 - 6 = 6 + D - 6 which makes 2 = D and also makes 2 = D = A
Solution
Substitute 3 for B, and 2 for D in eq.3a: E = 3 + 2 which makes E = 5 and makes ABCDEF = 233258