Puzzle for September 7, 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.
Scratchpad
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Hint #1
Add E to both sides of eq.4: D + E + E = C - E + E which becomes eq.4a) D + 2×E = C
Hint #2
In eq.5, replace C with D + 2×E (from eq.4a): D + 2×E + E = B + D which becomes D + 3×E = B + D Subtract D from both sides of the equation above: D + 3×E - D = B + D - D which makes 3×E = B
Hint #3
In eq.2, replace A with E + F (from eq.3): D = E + F + F which becomes eq.2a) D = E + 2×F
Hint #4
In eq.4a, substitute E + 2×F for D (from eq.2a): E + 2×F + 2×E = C which becomes eq.4b) 3×E + 2×F = C
Hint #5
Substitute 3×E for B, 3×E + 2×F for C (from eq.4b), E + F for A (from eq.3), and E + 2×F for D (from eq.2a) into eq.6: 3×E + 3×E + 2×F = E + F + E + 2×F + E which becomes 6×E + 2×F = 3×E + 3×F Subtract 2×F and 3×E from each side of the equation above: 6×E + 2×F - 2×F - 3×E = 3×E + 3×F - 2×F - 3×E which makes 3×E = F
Hint #6
Substitute 3×E for F in eq.3: A = E + 3×E which makes A = 4×E
Hint #7
Substitute (3×E) for F in eq.4b: 3×E + 2×(3×E) = C which becomes 3×E + 6×E = C which makes 9×E = C
Hint #8
Substitute (3×E) for F in eq.2a: D = E + 2×(3×E) which becomes D = E + 6×E which makes D = 7×E
Solution
Substitute 4×E for A, 3×E for B and F, 9×E for C, and 7×E for D in eq.1: 4×E + 3×E + 9×E + 7×E + E + 3×E = 27 which simplifies to 27×E = 27 Divide both sides of the above equation by 27: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 4×E = 4 × 1 = 4 B = F = 3×E = 3 × 1 = 3 C = 9×E = 9 × 1 = 9 D = 7×E = 7 × 1 = 7 and ABCDEF = 439713