Puzzle for September 13, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.6, replace D with B + F (from eq.2): A + B + F - F = B + E + F which becomes A + B = B + E + F Subtract B from each side of the above equation: A + B - B = B + E + F - B which becomes eq.6a) A = E + F
Hint #2
In eq.3, replace A with E + F (from eq.6a): F - E = E + F - F which becomes F - E = E Add E to both sides of the above equation: F - E + E = E + E which makes eq.3a) F = 2×E
Hint #3
In eq.4, substitute 2×E for F: B + E = 2×E - B In the equation above, subtract E from both sides, and add B to both sides: B + E - E + B = 2×E - B - E + B which makes 2×B = E
Hint #4
Substitute (2×B) for E in eq.2a: F = 2×(2×B) which makes F = 4×B
Hint #5
Substitute 2×B for E, and 4×B for F in eq.6a: A = 2×B + 4×B which becomes A = 6×B
Hint #6
Substitute 4×B for F in eq.2: D = B + 4×B which makes D = 5×B
Hint #7
Substitute 5×B for D, 4×B for F, and 6×B for A in eq.5: 5×B - 4×B = 6×B + C - 5×B which becomes B = B + C Subtract B from each side of the equation above: B - B = B + C - B which makes 0 = C
Solution
Substitute 6×B for A, 0 for C, 5×B for D, 2×B for E, and 4×B for F in eq.5: 6×B + B + 0 + 5×B + 2×B + 4×B = 18 which simplifies to 18×B = 18 Divide both sides of the above equation by 18: 18×B ÷ 18 = 18 ÷ 18 which means B = 1 making A = 6×B = 6 × 1 = 6 D = 5×B = 5 × 1 = 5 E = 2×B = 2 × 1 = 2 F = 4×B = 4 × 1 = 4 and ABCDEF = 610524