Puzzle for August 4, 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.3, replace C + F with B – C (from eq.4): B – C = A + E Add C to both sides of the above equation: B – C + C = A + E + C which becomes eq.3a) B = A + E + C
Hint #2
In eq.2, replace B with A + E + C (from eq.3a): A + E + C + E = A + D which becomes A + C + 2×E = A + D Subtract A from both sides of the equation above: A + C + 2×E – A = A + D – A which becomes eq.2a) C + 2×E = D
Hint #3
Substitute C + 2×E for D (from eq.2a) into eq.6: E + F = C + C + 2×E – E which becomes E + F = 2×C + E Subtract E from each side of the above equation: E + F – E = 2×C + E – E which makes F = 2×C
Hint #4
Substitute 2×C for F in eq.4: B – C = C + 2×C which becomes B – C = 3×C Add C to both sides of the above equation: B – C + C = 3×C + C which makes B = 4×C
Hint #5
Substitute 2×C for F in eq.5: A – C + 2×C = C + E which becomes A + C = C + E Subtract C from each side of the equation above: A + C – C = C + E – C which makes A = E
Hint #6
Substitute 2×C for F, and A for E in eq.3: C + 2×C = A + A which makes 3×C = 2×A Divide both sides of the above equation by 2: 3×C ÷ 2 = 2×A ÷ 2 which makes 1½×C = A and also makes E = A = 1½×C
Hint #7
Substitute 4×C for B, and A for E in eq.2: 4×C + A = A + D Subtract A from both sides of the equation above: 4×C + A – A = A + D – A which makes 4×C = D
Solution
Substitute 1½×C for A and E, 4×C for B and D, and 2×C for F in eq.1: 1½×C + 4×C + C + 4×C + 1½×C + 2×C = 28 which simplifies to 14×C = 28 Divide both sides of the above equation by 14: 14×C ÷ 14 = 28 ÷ 14 which means C = 2 making A = E = 1½×C = 1½ × 2 = 3 B = D = 4×C = 4 × 2 = 8 F = 2×C = 2 × 2 = 4 and ABCDEF = 382834