Puzzle for July 28, 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.5, replace C + D with A + F (from eq.3): B + A + F = A + E Subtract A from each side of the equation above: B + A + F – A = A + E – A which becomes eq.5a) B + F = E
Hint #2
In eq.4, replace E with B + F (from eq.5a): B + F + F = B + C which becomes B + 2×F = B + C Subtract B from each side of the above equation: B + 2×F – B = B + C – B which makes 2×F = C
Hint #3
eq.6 may be written as: B = (A + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × B = 3 × (A + D + F) ÷ 3 which becomes eq.6a) 3×B = A + D + F
Hint #4
In eq.5, substitute E + F for B + C (from eq.4): E + F + D = A + E Subtract E from both sides of the equation above: E + F + D – E = A + E – E which becomes F + D = A which is the same as eq.5b) D + F = A
Hint #5
Substitute A for D + F (from eq.5b) into eq.6a: 3×B = A + A which becomes 3×B = 2×A Divide both sides of the above equation by 2: 3×B ÷ 2 = 2×A ÷ 2 which makes eq.6b) 1½×B = A
Hint #6
Substitute 1½×B for A (from eq.6b), 2×F for C, and B + F for E (from eq.5a) in eq.2: 1½×B = 2×F + B + F which becomes 1½×B = 3×F + B Subtract B from each side of the above equation: 1½×B – B = 3×F + B – B which becomes ½×B = 3×F Multiply both sides by 2: 2×(½×B) = 2×(3×F) which makes B = 6×F
Hint #7
Substitute (6×F) for B in eq.6b: 1½×(6×F) = A which makes 9×F = A
Hint #8
Substitute 6×F for B in eq.5a: 6×F + F = E which makes 7×F = E
Hint #9
Substitute 9×F for A in eq.5b: D + F = 9×F Subtract F from each side of the above equation: D + F – F = 9×F – F which makes D = 8×F
Solution
Substitute 9×F for A, 6×F for B, 2×F for C, 8×F for D, and 7×F for E in eq.1: 9×F + 6×F + 2×F + 8×F + 7×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 = 9×F = 9 × 1 = 9 B = 6×F = 6 × 1 = 6 C = 2×F = 2 × 1 = 2 D = 8×F = 8 × 1 = 8 E = 7×F = 7 × 1 = 7 and ABCDEF = 962871