Puzzle for August 30, 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.4, replace F with D + E (from eq.2): E + D + E = C + D which becomes D + 2×E = C + D Subtract D from both sides of the above equation: D + 2×E – D = C + D – D which becomes 2×E = C
Hint #2
In eq.3, replace C with 2×E: eq.3a) 2×E = A + B In eq.1, substitute 2×E for A + B (from eq.3a), 2×E for C, and D + E for F (from eq.2): 2×E + 2×E + D + E + D + E = 34 which becomes eq.1a) 6×E + 2×D = 34
Hint #3
eq.1 may be written as: A + C + B + D + F + E = 34 In the above equation, substitute A + C for B + D + F (from eq.5): A + C + A + C + E = 34 which becomes eq.1b) 2×A + 2×C + E = 34
Hint #4
Substitute (2×E) for C in eq.1b: 2×A + 2×(2×E) + E = 34 which is equivalent to 2×A + 4×E + E = 34 which becomes eq.1c) 2×A + 5×E = 34
Hint #5
In eq.1c, substitute 6×E + 2×D for 34 (from eq.1a): 2×A + 5×E = 6×E + 2×D Subtract 5×E from both sides of the equation above: 2×A + 5×E – 5×E = 6×E + 2×D – 5×E which becomes 2×A = E + 2×D Divide both sides by 2: 2×A ÷ 2 = (E + 2×D) ÷ 2 which makes eq.1d) A = ½×E + D
Hint #6
Subtract E from both sides of eq.6: D – E = B + E – E which becomes eq.6a) D – E = B Substitute 2×E for C, ½×E + D for A (from eq.1c), and D – E for B (from eq.6a) in eq.3: 2×E = ½×E + D + D – E which becomes 2×E = 2×D – ½×E Add ½×E to both sides of the above equation: 2×E + ½×E = 2×D – ½×E + ½×E which makes 2½×E = 2×D Divide both sides by 2: 2½×E ÷ 2 = 2×D ÷ 2 which makes 1¼×E = D
Hint #7
Substitute 1¼×E for D in eq.1d: A = ½×E + 1¼×E which makes A = 1¾×E
Hint #8
Substitute 1¼×E for D in eq.6a: 1¼×E – E = B which makes ¼×E = B
Hint #9
Substitute 1¼×E for D in eq.2: F = 1¼×E + E which makes F = 2¼×E
Solution
Substitute (1¾×E) for A in eq.1c: 2×(1¾×E) + 5×E = 34 which becomes 3½×E + 5×E = 34 which makes 8½×E = 34 Divide both sides by 8½: 8½×E ÷ 8½ = 34 ÷ 8½ which means E = 4 making A = 1¾×E = 1¾ × 4 = 7 B = ¼×E = ¼ × 4 = 1 C = 2×E = 2 × 4 = 8 D = 1¼×E = 1¼ × 4 = 5 F = 2¼×E = 2¼ × 4 = 9 and ABCDEF = 718549