Puzzle for September 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.3, replace B with A + E (from eq.2): A + C = A + E + E which becomes A + C = A + 2×E Subtract A from both sides of the above equation: A + C – A = A + 2×E – A which makes C = 2×E
Hint #2
In eq.4, replace C with 2×E: D + E = 2×E – E which becomes D + E = E Subtract E from both sides of the equation above: D + E – E = E – E which makes D = 0
Hint #3
In eq.5, substitute A + E for B (from eq.2), and 2×E for C: A + E + 2×E – F = A + F which becomes A + 3×E – F = A + F In the equation above, subtract A from both sides, and add F to each side: A + 3×E – F – A + F = A + F – A + F which simplifies to 3×E = 2×F Divide both sides by 2: 3×E ÷ 2 = 2×F ÷ 2 which makes 1½×E = F
Hint #4
Substitute 2×E for C, and 1½×E for F in eq.6: 2×E + 1½×E = A which makes 3½×E = A
Hint #5
Substitute 3½×E for A in eq.2: B = 3½×E + E which makes B = 4½×E
Solution
Substitute 3½×E for A, 4½×E for B, 2×E for C, 0 for D, and 1½×E for F in eq.1: 3½×E + 4½×E + 2×E + 0 + E + 1½×E = 25 which simplifies to 12½×E = 25 Divide both sides of the equation above by 12½: 12½×E ÷ 12½ = 25 ÷ 12½ which means E = 2 making A = 3½×E = 3½ × 2 = 7 B = 4½×E = 4½ × 2 = 9 C = 2×E = 2 × 2 = 4 F = 1½×E = 1½ × 2 = 3 and ABCDEF = 794023