Puzzle for October 30, 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
In eq.6, replace F with B + D + E (from eq.5), and replace A with B + C – D (from eq.2): C + B + D + E = B + C – D + E Subtract B, C, and E from both sides of the above equation: C + B + D + E – B – C – E = B + C – D + E – B – C – E which simplifies to D = –D Add D to both sides: D + D = –D + D which becomes 2×D = 0 which means D = 0
Hint #2
In eq.5, replace D with 0: B + 0 + E = F which makes B + E = F In eq.4, replace F with (B + E): E – (B + E) = B – C which is equivalent to E – B – E = B – C which becomes –B = B – C Add B and C to both sides of the above equation: –B + B + C = B – C + B + C which makes C = 2×B
Hint #3
In eq.2, substitute 2×B for C, and 0 for D: A = B + 2×B – 0 which makes A = 3×B
Hint #4
Substitute 3×B for A, and 2×B for C in eq.3: F = 3×B + B + 2×B which makes F = 6×B
Hint #5
Substitute 2×B for C, 6×B for F, and 3×B for A in eq.6: 2×B + 6×B = 3×B + E Subtract 3×B from both sides: 2×B + 6×B – 3×B = 3×B + E – 3×B which makes 5×B = E
Solution
Substitute 3×B for A, 2×B for C, 0 for D, 5×B for E, and 6×B for F in eq.1: 3×B + B + 2×B + 0 + 5×B + 6×B = 17 which simplifies to 17×B = 17 Divide both sides by 17: 17×B ÷ 17 = 17 ÷ 17 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 2×B = 2 × 1 = 2 E = 5×B = 5 × 1 = 5 F = 6×B = 6 × 1 = 6 and ABCDEF = 312056