Puzzle for March 16, 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.4, replace D + E with B + C + F (from eq.3): C + B + C + F = A + B + F which becomes 2×C + B + F = A + B + F Subtract both B and F from each side of the equation above: 2×C + B + F – B – F = A + B + F – B – F which simplifies to 2×C = A
Hint #2
In eq.2, replace A with 2×C: C + F = 2×C Subtract C from both sides of the above equation: C + F – C = 2×C – C which makes F = C
Hint #3
In eq.5, substitute C for F, and 2×C for A: B + C = 2×C + C – C which becomes B + C = 2×C Subtract C from both sides of the equation above: B + C – C = 2×C – C which makes B = C
Hint #4
Substitute 2×C for A, and C for B in eq.6: 2×C – E = C + E In the above equation, add E to each side, and subtract C from both sides: 2×C – E + E – C = C + E + E – C which makes C = 2×E Divide both sides by 2: C ÷ 2 = 2×E ÷ 2 which makes ½×C = E
Hint #5
Substitute ½×C for E, and C for B and F in eq.3: D + ½×C = C + C + C D + ½×C = 3×C Subtract ½×C from each side of the above equation: D + ½×C – ½×C = 3×C – ½×C which makes D = 2½×C
Solution
Substitute 2×C for A, C for B and F, 2½×C for D, and ½×C for E in eq.1: 2×C + C + C + 2½×C + ½×C + C = 16 which simplifies to 8×C = 16 Divide each side of the equation above by 8: 8×C ÷ 8 = 16 ÷ 8 which means C = 2 making A = 2×C = 2 × 2 = 4 B = F = C = 2 D = 2½×C = 2½ × 2 = 5 E = ½×C = ½ × 2 = 1 and ABCDEF = 422512