Puzzle for June 28, 2019 ( )
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
eq.2 may be re-written as: F = B + A + C In the above equation, replace A + C with D + E (from eq.3): eq.2a) F = B + D + E
Hint #2
In eq.5, replace F with B + D + E (from eq.2a): D + B + D + E = A + B + E which becomes B + 2×D + E = A + B + E Subtract B and E from both sides of the above equation: B + 2×D + E - B - E = A + B + E - B - E which simplifies to 2×D = A
Hint #3
In eq.3, replace A with 2×D: 2×D + C = D + E Subtract D from each side of the above equation: 2×D + C - D = D + E - D which becomes eq.3a) D + C = E
Hint #4
In eq.4, substitute D + C for E (from eq.3a): D + C = B + C Subtract C from each side of the equation above: D + C - C = B + C - C which makes D = B
Hint #5
Substitute 2×D for A, and D for B in eq.2: F = 2×D + D + C which becomes eq.2b) F = 3×D + C
Hint #6
Substitute D for B, and 3×D + C for F (from eq.2b) in eq.6: C + E = D + D + 3×D + C which becomes C + E = 5×D + C Subtract C from both sides: C + E - C = 5×D + C - C which makes E = 5×D
Hint #7
Substitute 5×D for E in eq.3a: D + C = 5×D Subtract D from each side: D + C - D = 5×D - D which makes C = 4×D
Hint #8
Substitute 4×D for C in eq.2b: F = 3×D + 4×D which makes F = 7×D
Solution
Substitute 2×D for A, D for B, 4×D for C, 5×D for E, and 7×D for F in eq.1: 2×D + D + 4×D + D + 5×D + 7×D = 20 which becomes 20×D = 20 Divide both sides of the above equation by 20: 20×D ÷ 20 = 20 ÷ 20 which means D = 1 making A = 2×D = 2 × 1 = 2 B = D = 1 C = 4×D = 4 × 1 = 4 E = 5×D = 5 × 1 = 5 F = 7×D = 7 × 1 = 7 and ABCDEF = 214157