Puzzle for October 12, 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.5, replace C with B + D (from eq.2): B + D – D = B – F which becomes B = B – F Subtract B from each side of the equation above: B – B = B – F – B which makes 0 = –F which means 0 = F
Hint #2
eq.3 may be written as: B + E = C – B + E In the equation above, subtract E from each side, and add B to each side: B + E – E + B = C – B + E – E + B which makes 2×B = C
Hint #3
In eq.2, replace C with 2×B: 2×B = B + D Subtract B from each side of the equation above: 2×B – B = B + D – B which makes B = D
Hint #4
In eq.4, substitute 2×B for C, and 0 for F: A + B = 2×B + 0 which becomes A + B = 2×B Subtract B from both sides of the equation above: A + B – B = 2×B – B which makes A = B and means A = B = D
Hint #5
Substitute B for D and A, and 2×B for C in eq.6: B – E = (B + B) ÷ 2×B which becomes B – E = (2×B) ÷ 2×B which makes B – E = 1 In the above equation, subtract 1 from both sides, and add E to both sides: B – E – 1 + E = 1 – 1 + E which makes eq.6a) B – 1 = E
Solution
Substitute B for A and D, 2×B for C, B – 1 for E (from eq.6a), and 0 for F in eq.1: B + B + 2×B + B + B – 1 + 0 = 23 which simplifies to 6×B – 1 = 23 Add 1 to both sides of the above equation: 6×B – 1 + 1 = 23 + 1 which makes 6×B = 24 Divide both sides by 6: 6×B ÷ 6 = 24 ÷ 6 which means B = 4 making A = D = B = 4 C = 2×B = 2 × 4 = 8 E = B – 1 = 4 – 1 = 3 (from eq.6a) and ABCDEF = 448430