Puzzle for March 18, 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.3, replace E with D + F (from eq.4): B = D + F + F which becomes eq.3a) B = D + 2×F
Hint #2
In eq.6, replace B with D + 2×F (from eq.3a): D + E = D + 2×F + C + F which becomes D + E = D + 3×F + C Subtract D from each side of the above equation: D + E – D = D + 3×F + C – D which becomes eq.6a) E = 3×F + C
Hint #3
In eq.2, substitute 3×F + C for E (from eq.6a): 3×F + C = A + C Subtract C from both sides of the equation above: 3×F + C – C = A + C – C which makes 3×F = A
Hint #4
Subtract A and D from both sides of eq.5: A + B – A – D = C + D – A – D which becomes eq.5a) B – D = C – A Subtract D from both sides of eq.6: D + E – D = B + C + F – D which becomes E = B + C + F – D which may be written as eq.6b) E = B – D + C + F
Hint #5
Substitute A + C for E (from eq.2), and C – A for B – D (from eq.5a) into eq.6b: A + C = C – A + C + F which becomes A + C = 2×C – A + F In the above equation, subtract C from both sides, and add A to both sides: A + C – C + A = 2×C – A + F – C + A which becomes eq.6c) 2×A = C + F
Hint #6
Substitute (3×F) for A in eq.6c: 2×(3×F) = C + F which becomes 6×F = C + F Subtract F from each side of the above equation: 6×F – F = C + F – F which makes 5×F = C
Hint #7
Substitute 3×F for A, and 5×F for C in eq.2: E = 3×F + 5×F which makes E = 8×F
Hint #8
Substitute 8×F for E in eq.4: 8×F = D + F Subtract F from each side of the equation above: 8×F – F = D + F – F which makes 7×F = D
Hint #9
Substitute 8×F for E in eq.3: B = 8×F + F which makes B = 9×F
Solution
Substitute 3×F for A, 9×F for B, 5×F for C, 7×F for D, and 8×F for E in eq.1: 3×F + 9×F + 5×F + 7×F + 8×F + F = 33 which simplifies to 33×F = 33 Divide both sides of the above equation by 33: 33×F ÷ 33 = 33 ÷ 33 which means F = 1 making A = 3×F = 3 × 1 = 3 B = 9×F = 9 × 1 = 9 C = 5×F = 5 × 1 = 5 D = 7×F = 7 × 1 = 7 E = 8×F = 8 × 1 = 8 and ABCDEF = 395781