Puzzle for August 4, 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.2: E - F + E = D - E + E which becomes eq.2a) 2×E - F = D
Hint #2
In eq.3, replace D with 2×E - F (from eq.2a): 2×E - F + F = C - F which becomes 2×E = C - F Add F to both sides of the above equation: 2×E + F = C - F + F which becomes eq.3a) 2×E + F = C
Hint #3
In eq.5, substitute 2×E + F for C (from eq.3a), and 2×E - F for D (from eq.2a): 2×E + F + E + F = A + 2×E - F which becomes 3×E + 2×F = A + 2×E - F In the equation above, subtract 2×E from both sides, and add F to both sides: 3×E + 2×F - 2×E + F = A + 2×E - F - 2×E + F which becomes eq.5a) E + 3×F = A
Hint #4
Substitute (2×E + F) for C (from eq.3a), and E + 3×F for A (from eq.5a) into eq.4: B - (2×E + F) = E + 3×F - B which becomes B - 2×E - F = E + 3×F - B Add 2×E, F, and B to both sides of the above equation: B - 2×E - F + 2×E + F + B = E + 3×F - B + 2×E + F + B which simplifies to 2×B = 3×E + 4×F Divide both sides by 2: 2×B ÷ 2 = (3×E + 4×F) ÷ 2 which becomes eq.4a) B = 1½×E + 2×F
Hint #5
eq.6 may be written as: A = (B + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (B + C + E) ÷ 3 which becomes eq.6a) 3×A = B + C + E
Hint #6
Substitute (E + 3×F) for A (from eq.5a), 1½×E + 2×F for B (from eq.4a), and 2×E + F for C (from eq.3a) in eq.6a: 3×(E + 3×F) = 1½×E + 2×F + 2×E + F + E which becomes 3×E + 9×F = 4½×E + 3×F Subtract 3×E and 3×F from each side of the equation above: 3×E + 9×F - 3×E - 3×F = 4½×E + 3×F - 3×E - 3×F which becomes 6×F = 1½×E Divide both sides by 1½: 6×F ÷ 1½ = 1½×E ÷ 1½ which makes 4×F = E
Hint #7
Substitute 4×F for E in eq.5a: 4×F + 3×F = A which makes 7×F = A
Hint #8
Substitute (4×F) for E in eq.4a: B = 1½×(4×F) + 2×F which becomes B = 6×F + 2×F which makes B = 8×F
Hint #9
Substitute (4×F) for E in eq.3a: 2×(4×F) + F = C which becomes 8×F + F = C which makes 9×F = C
Hint #10
Substitute (4×F) for E in eq.2a: 2×(4×F) - F = D which becomes 8×F - F = D which makes 7×F = D
Solution
Substitute 7×F for A and D, 8×F for B, 9×F for C, and 4×F for E in eq.1: 7×F + 8×F + 9×F + 7×F + 4×F + F = 36 which simplifies to 36×F = 36 Divide both sides of the above equation by 36: 36×F ÷ 36 = 36 ÷ 36 which means F = 1 making A = D = 7×F = 7 × 1 = 7 B = 8×F = 8 × 1 = 8 C = 9×F = 9 × 1 = 9 E = 4×F = 4 × 1 = 4 and ABCDEF = 789741