Puzzle for April 18, 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
eq.1 may be re-written as: A + C + B + D + E + F = 26 Replace B + D + E + F with A + C (from eq.6) in the above equation: A + C + A + C = 26 which becomes 2×A + 2×C = 26 Divide both sides by 2: (2×A + 2×C) ÷ 2 = 26 ÷ 2 which makes eq.1a) A + C = 13
Hint #2
In eq.6, replace A + C with 13 (from eq.1a): B + D + E + F = 13 Replace D + E with F (from eq.3): B + F + F = 13 which becomes eq.6a) B + 2×F = 13
Hint #3
In eq.5, substitute A + E for C + D (from eq.4): A + F = B + A + E Subtract A from both sides of the above equation: A + F - A = B + A + E - A which becomes eq.5a) F = B + E
Hint #4
In eq.5a, substitute D + E for F (from eq.3): D + E = B + E Subtract E from both sides of the above equation: D + E - E = B + E - E which makes D = B
Hint #5
Substitute B for D in eq.2: B + B = C which becomes 2×B = C
Hint #6
Substitute 2×B for C in eq.1a: A + 2×B = 13 Subtract 2×B from each side: A + 2×B - 2×B = 13 - 2×B which means eq.1b) A = 13 - 2×B
Hint #7
Subtract B from both sides of eq.6a: B + 2×F - B = 13 - B which becomes 2×F = 13 - B Divide both sides by 2: 2×F ÷ 2 = (13 - B) ÷ 2 which makes eq.6b) F = 6½ - ½×B
Hint #8
Substitute 2×B for C, B for D, and 13 - 2×B for A (from eq.1b) in eq.4: 2×B + B = 13 - 2×B + E Add 2×B - 13 to both sides of the above equation: 2×B + B + 2×B - 13 = 13 - 2×B + E + 2×B - 13 which simplifies to eq.4a) 5×B - 13 = E
Solution
Substitute 13 - 2×B for A (from eq.1b), 2×B for C, B for D, 5×B - 13 for E (from eq.4a), and 6½ - ½×B for F (from eq.6b) in eq.1: 13 - 2×B + B + 2×B + B + 5×B - 13 + 6½ - ½×B = 26 which simplifies to 6½×B + 6½ = 26 Subtract 6½ from both sides: 6½×B + 6½ - 6½ = 26 - 6½ which makes 6½×B = 19½ Divide both sides by 6½: 6½×B ÷ 6½ = 19½ ÷ 6½ which means B = 3 making A = 13 - 2×B = 13 - 2×3 = 13 - 6 = 7 (from eq.1b) C = 2×B = 2×3 = 6 D = B = 3 E = 5×B - 13 = 5×3 - 13 = 15 - 13 = 2 (from eq.4a) F = 6½ - ½×B = 6½ - ½×3 = 6½ - 1½ = 5 (from eq.6b) and ABCDEF = 736325