Puzzle for August 23, 2021 ( )
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.4, replace B + E with D (from eq.2): C + D = D + F Subtract D from each side of the equation above: C + D – D = D + F – D which makes C = F
Hint #2
In eq.3, replace F with C: C = C + E Subtract C from each side of the above equation: C – C = C + E – C which makes 0 = E
Hint #3
In eq.2, substitute 0 for E: D = B + 0 which makes D = B
Hint #4
Substitute B for D, 0 for E, and C for F in eq.5: B + B + 0 – C = C – 0 + C which becomes 2×B – C = 2×C Add C to both sides of the above equation: 2×B – C + C = 2×C + C which makes 2×B = 3×C Divide both sides by 2: 2×B ÷ 2 = 3×C ÷ 2 which makes B = 1½×C and also makes D = B = 1½×C
Hint #5
Substitute 0 for E, B for D, and C for F in eq.6: A – C + 0 = (B ÷ B) + (C ÷ C) which becomes A – C = 1 + 1 which means A – C = 2 Add C to both sides of the equation above: A – C + C = 2 + C which makes eq.6a) A = 2 + C
Solution
Substitute 2 + C for A (from eq.6a), 1½×C for B and D, 0 for E, and C for F in eq.1: 2 + C + 1½×C + C + 1½×C + 0 + C = 38 which simplifies to 2 + 6×C = 38 Subtract 2 from each side of the above equation: 2 + 6×C – 2 = 38 – 2 which makes 6×C = 36 Divide both sides by 6: 6×C ÷ 6 = 36 ÷ 6 which means C = 6 making A = 2 + C = 2 + 6 = 8 (from eq.6a) B = D = 1½×C = 1½ × 6 = 9 F = C = 6 and ABCDEF = 896906