Puzzle for June 2, 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 B + E with D + F (from eq.2): C + F = A + D + F Subtract F from both sides of the above equation: C + F – F = A + D + F – F which becomes eq.3a) C = A + D
Hint #2
In eq.4, replace C with A + D (from eq.3a): A + F = A + D + D which becomes A + F = A + 2×D Subtract A from both sides of the equation above: A + F – A = A + 2×D – A which makes F = 2×D
Hint #3
In eq.5, substitute A + D for C (from eq.3a), and 2×D for F: A + D + D + E = A + B – D + 2×D which becomes A + 2×D + E = A + B + D Subtract A and D from both sides of the equation above: A + 2×D + E – A – D = A + B + D – A – D which simplifies to eq.5a) D + E = B
Hint #4
Substitute D + E for B (from eq.5a), and 2×D for F in eq.2: D + E + E = D + 2×D which becomes D + 2×E = 3×D Subtract D from each side of the above equation: D + 2×E – D = 3×D – D which makes 2×E = 2×D Divide both sides by 2: 2×E ÷ 2 = 2×D ÷ 2 which makes E = D
Hint #5
Substitute D for E in eq.5a: D + D = B which makes 2×D = B
Hint #6
Substitute 2×D for B and F in eq.6: 2×D – C + 2×D – (A + C) = C – 2×D which becomes 4×D – C – A – C = C – 2×D which becomes 4×D – A – 2×C = C – 2×D Add 2×C and 2×D to both sides of the above equation: 4×D – A – 2×C + 2×C + 2×D = C – 2×D + 2×C + 2×D which makes eq.6a) 6×D – A = 3×C
Hint #7
Substitute (A + D) for C (from eq.3a) in eq.6a: 6×D – A = 3×(A + D) which becomes 6×D – A = 3×A + 3×D In the above equation, add A to both sides, and subtract 3×D from both sides: 6×D – A + A – 3×D = 3×A + 3×D + A – 3×D which becomes 3×D = 4×A Divide each side by 4: 3×D ÷ 4 = 4×A ÷ 4 which makes ¾×D = A
Hint #8
Substitute ¾×D for A in eq.3a: C = ¾×D + D which makes C = 1¾×D
Solution
Substitute ¾×D for A, 2×D for B and F, 1¾×D for C, and D for E in eq.1: ¾×D + 2×D + 1¾×D + D + D + 2×D = 34 which simplifies to 8½×D = 34 Divide both sides of the equation above by 8½: 8½×D ÷ 8½ = 34 ÷ 8½ which means D = 4 making A = ¾×D = ¾ × 4 = 3 B = F = 2×D = 2 × 4 = 8 C = 1¾×D = 1¾ × 4 = 7 E = D = 4 and ABCDEF = 387448