Puzzle for November 12, 2020 ( )
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
Subtract D from both sides of eq.4: D – F – D = C – B – D which becomes –F = C – B – D which may be written as –F = C – D – B In the above equation, replace C – D with F (from eq.2): –F = F – B Add F and B to both sides: –F + F + B = F – B + F + B which means B = 2×F
Hint #2
In eq.3, replace B with 2×F: C – E = A – 2×F Add 2×F to both sides of the equation above: C – E + 2×F = A – 2×F + 2×F which becomes eq.3a) C – E + 2×F = A
Hint #3
In eq.5, substitute C – E + 2×F for A (from eq.3a): E + F = C – E + 2×F + D In the equation above, subtract F from each side, and add E to each side: E + F – F + E = C – E + 2×F + D – F + E which becomes eq.5a) 2×E = C + F + D
Hint #4
Substitute C – D for F (from eq.2) in eq.5a: 2×E = C + C – D + D which becomes 2×E = 2×C Divide both sides of the above equation by 2: 2×E ÷ 2 = 2×C ÷ 2 which makes E = C
Hint #5
Substitute C for E in eq.3a: C – C + 2×F = A which makes 2×F = A
Hint #6
Add D to both sides of eq.2: F + D = C – D + D which becomes eq.2a) F + D = C Substitute 2×F for A and B, and (F + D) for C (from eq.2a) in eq.6: 2×F + 2×F – (F + D) = D – 2×F – 2×F which becomes 4×F – F – D = D – 4×F which becomes 3×F – D = D – 4×F Add both D and 4×F to each side of the equation above: 3×F – D + D + 4×F = D – 4×F + D + 4×F which simplifies to 7×F = 2×D Divide both sides by 2: 7×F ÷ 2 = 2×D ÷ 2 which makes 3½×F = D
Hint #7
Substitute 3½×F for D in eq.2a: F + 3½×F = C which makes 4½×F = C and also makes E = C = 4½×F
Solution
Substitute 2×F for A and B, 4½×F for C and E, and 3½×F for D in eq.1: 2×F + 2×F + 4½×F + 3½×F + 4½×F + F = 35 which simplifies to 17½×F = 35 Divide both sides of the equation above by 17½: 17½×F ÷ 17½ = 35 ÷ 17½ which means F = 2 making A = B = 2×F = 2 × 2 = 4 C = E = 4½×F = 4½ × 2 = 9 D = 3½×F = 3½ × 2 = 7 and ABCDEF = 449792