Puzzle for July 27, 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
Add B and C to both sides of eq.3: D - B - C + B + C = A + B + C which becomes eq.3a) D = A + B + C In eq.2, replace A + B + C with D: eq.3b) E + F = D
Hint #2
In eq.1, substitute D for A + B + C (from eq.3a) and for E + F (from eq.3b): D + D + D = 27 which makes 3×D = 27 Divide both sides of the above equation by 3: 3×D ÷ 3 = 27 ÷ 3 which makes D = 9
Hint #3
Add A and E to both sides of eq.5: F - A + C + A + E = B - E + A + E which becomes F + C + E = B + A which may be written as eq.5a) E + F + C = A + B Substitute A + B + C for E + F (from eq.2) in eq.5a: A + B + C + C = A + B which becomes A + B + 2×C = A + B Subtract A and B from both sides: A + B + 2×C - A - B = A + B - A - B which simplifies to 2×C = 0 which means C = 0
Hint #4
Substitute 9 for D, and 0 for C in eq.3: 9 - B - 0 = A which makes eq.3c) 9 - B = A
Hint #5
Add B to both sides of eq.4: B + F + B = A - B + B which becomes eq.4a) 2×B + F = A Substitute 9 - B for A (from eq.3c) in eq.4a: 2×B + F = 9 - B Subtract 2×B from both sides: 2×B + F - 2×B = 9 - B - 2×B which becomes eq.4b) F = 9 - 3×B
Hint #6
Substitute 0 for C, and 2×B + F for A (from eq.4a) in eq.5a: E + F + 0 = 2×B + F + B which becomes E + F = 3×B + F Subtract F from both sides: E + F - F = 3×B + F - F which makes E = 3×B
Solution
Substitute 9 for D, 3×B for E, 9 - 3×B for F (from eq.4b), 9 - B for A (from eq.3c), and 3×B for E in eq.6: 9 - 3×B + 9 - 3×B = 9 - B + B + 3×B which becomes 18 - 6×B = 9 + 3×B In the above equation, add 6×B to both sides, and subtract 9 from each side: 18 - 6×B + 6×B - 9 = 9 + 3×B + 6×B - 9 which makes 9 = 9×B Divide both sides by 9: 9 ÷ 9 = 9×B ÷ 9 which means 1 = B making A = 9 - B = 9 - 1 = 8 (from eq.3c) E = 3×B = 3×1 = 3 F = 9 - 3×B = 9 - 3×1 = 9 - 3 = 6 (from eq.4b) and ABCDEF = 810936