Puzzle for June 15, 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.
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Hint #1
Add C to both sides of eq.3: C + E + C = F – C + C which becomes eq.3a) 2×C + E = F In eq.2, replace F with 2×C + E (from eq.3a): 2×C + E = A + E Subtract E from each side of the equation above: 2×C + E – E = A + E – E which makes 2×C = A
Hint #2
In eq.4, replace A with 2×C: 2×C + B – C = C – B which becomes C + B = C – B In the above equation, subtract C from both sides, and add B to both sides: C + B – C + B = C – B – C + B which makes 2×B = 0 which means B = 0
Hint #3
In eq.5, substitute 2×C + E for F (from eq.3a), and 0 for B: D – E + 2×C + E = E + 0 which becomes eq.5a) D + 2×C = E
Hint #4
Substitute D + 2×C for E (from eq.5a) into eq.3a: 2×C + D + 2×C = F which becomes eq.3b) 4×C + D = F
Hint #5
Substitute 4×C + D for F (from eq.3b) in eq.6: 4×C + D – C – D = C + D which becomes 3×C = C + D Subtract C from each side of the equation above: 3×C – C = C + D – C which makes 2×C = D
Hint #6
Substitute 2×C for D in eq.5a: 2×C + 2×C = E which makes 4×C = E
Hint #7
Substitute 2×C for D in eq.3b: 4×C + 2×C = F which makes 6×C = F
Solution
Substitute 2×C for A and D, 0 for B, 4×C for E, and 6×C for F in eq.1: 2×C + 0 + C + 2×C + 4×C + 6×C = 15 which simplifies to 15×C = 15 Divide both sides of the above equation by 15: 15×C ÷ 15 = 15 ÷ 15 which means C = 1 making A = D = 2×C = 2×1 = 2 E = 4×C = 4×1 = 4 F = 6×C = 6×1 = 6 and ABCDEF = 201246