Puzzle for June 18, 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
In eq.3, replace A with E + F (from eq.4): E + F – C = B + F In the above equation, subtract F from each side, and add C to each side: E + F – C – F + C = B + F – F + C which becomes eq.3a) E = B + C In eq.2, replace B + C with E (from eq.3a): D = E
Hint #2
In eq.5, add F to each side, and subtract A from each side: C + D – F + F – A = A + B + F – A which becomes C + D – A = B + F In eq.3, substitute C + D – A for B + F: A – C = C + D – A Add A and C to both sides of the equation above: A – C + A + C = C + D – A + A + C which becomes eq.3b) 2×A = 2×C + D
Hint #3
Add D to both sides of eq.6: B + D + D = A + C – D + D which becomes B + 2×D = A + C Substitute (B + C) for D (from eq.2) in the equation above: B + 2×(B + C) = A + C which is equivalent to B + 2×B + 2×C = A + C which becomes 3×B + 2×C = A + C Subtract C from both sides of the above equation: 3×B + 2×C – C = A + C – C which becomes eq.6a) 3×B + C = A
Hint #4
Substitute (3×B + C) for A (from eq.6a) in eq.3b: 2×(3×B + C) = 2×C + D which becomes 6×B + 2×C = 2×C + D Subtract 2×C from each side of the above equation: 6×B + 2×C – 2×C = 2×C + D – 2×C which makes 6×B = D and also makes E = D = 6×B
Hint #5
Substitute 6×B for E in eq.3a: 6×B = B + C Subtract B from both sides of the equation above: 6×B – B = B + C – B which makes 5×B = C
Hint #6
Substitute 5×B for C in eq.6a: 3×B + 5×B = A which makes 8×B = A
Hint #7
Substitute 6×B for E, and 8×B for A in eq.4: 6×B + F = 8×B Subtract 6×B from each side of the equation above: 6×B + F – 6×B = 8×B – 6×B which makes F = 2×B
Solution
Substitute 8×B for A, 5×B for C, 6×B for D and E, and 2×B for F in eq.1: 8×B + B + 5×B + 6×B + 6×B + 2×B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 5×B = 5 × 1 = 5 D = E = 6×B = 6 × 1 = 6 F = 2×B = 2 × 1 = 2 and ABCDEF = 815662