Puzzle for January 25, 2023 ( )
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.
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Hint #1
In eq.5, replace F with B + E (from eq.2): C + B + E - B = B + D which becomes eq.5a) C + E = B + D
Hint #2
In eq.4, replace B + D with C + E (from eq.5a): A + E = C + E Subtract E from each side of the equation above: A + E - E = C + E - E which makes A = C
Hint #3
In eq.6, replace F with B + E (from eq.2): B + C = D + E + B + E which becomes B + C = D + 2×E + B Subtract B from each side of the above equation: B + C - B = D + 2×E + B - B which becomes eq.6a) C = D + 2×E
Hint #4
In eq.5a, substitute D + 2×E for C (from eq.6a): D + 2×E + E = B + D which becomes D + 3×E = B + D Subtract D from each side of the equation above: D + 3×E - D = B + D - D which makes 3×E = B
Hint #5
In eq.2, substitute 3×E for B: F = 3×E + E which makes F = 4×E
Hint #6
Substitute 3×E for B, 4×E for F, and A for C in eq.3: 3×E + 4×E = A + A which becomes 7×E = 2×A Divide both sides of the above equation by 2: 7×E ÷ 2 = 2×A ÷ 2 which makes 3½×E = A and also makes 3½×E = A = C
Hint #7
Substitute 3½×E for C in eq.6a: 3½×E = D + 2×E Subtract 2×E from each side of the above equation: 3½×E = D + 2×E which makes 1½×E = D
Solution
Substitute 3½×E for A and C, 3×E for B, 1½×E for D, and 4×E for F in eq.1: 3½×E + 3×E + 3½×E + 1½×E + E + 4×E = 33 which simplifies to 16½×E = 33 Divide both sides of the above equation by 16½: 16½×E ÷ 16½ = 33 ÷ 16½ which means E = 2 making A = C = 3½×E = 3½ × 2 = 7 B = 3×E = 3 × 2 = 3 D = 1½×E = 1½ × 2 = 2 F = 4×E = 4 × 2 = 8 and ABCDEF = 767328