Puzzle for October 5, 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.3, replace E + F with A (from eq.2): B – A = C – (A) which is the same as B – A = C – A Add A to both sides of the above equation: B – A + A = C – A + A which makes B = C
Hint #2
In eq.4, replace C with B: B = B + D Subtract B from each side of the above equation: B – B = B + D – B which makes 0 = D
Hint #3
In eq.5, substitute 0 for D: F = 0 + E which makes F = E
Hint #4
Substitute E for F in eq.2: E + E = A which makes 2×E = A
Hint #5
Substitute 2×E for A, E for F, and B for C in eq.6: 2×E + E + E = B – 2×E + B – E which becomes 4×E = 2×B – 3×E Add 3×E to both sides of the equation above: 4×E + 3×E = 2×B – 3×E + 3×E which makes 7×E = 2×B Divide each side by 2: 7×E ÷ 2 = 2×B ÷ 2 which makes 3½×E = B and also makes C = B = 3½×E
Solution
Substitute 3½×E for B and C, 2×E for A, 0 for D, and E for F in eq.1: 3½×E + 2×E + 0 + 3½×E + E + E = 11×E which simplifies to 11×E = 22 Divide both sides of the above equation by 11: 11×E ÷ 11 = 22 ÷ 11 which means E = 2 making B = C = 3½×E = 3½ × 2 = 7 A = 2×E = 2 × 2 = 4 F = E = 2 and ABCDEF = 477022