Puzzle for July 19, 2022 ( )
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 A with B + F (from eq.2): B + D = B + F + F which becomes B + D = B + 2×F Subtract B from each side of the equation above: B + D – B = B + 2×F – B which makes eq.4a) D = 2×F
Hint #2
In eq.3, replace D with 2×F: E + F = 2×F – E In the above equation, subtract F from both sides, and add E to both sides: E + F – F + E = 2×F – E – F + E which makes 2×E = F
Hint #3
In eq.4a, substitute 2×E for F: D = 2×(2×E) which makes D = 4×E
Hint #4
In eq.5, substitute 2×E for F: 2×E – B = B – E Add B and E to both sides of the above equation: 2×E – B + B + E = B – E + B + E which becomes 3×E = 2×B Divide both sides by 2: 3×E ÷ 2 = 2×B ÷ 2 which makes 1½×E = B
Hint #5
Substitute 1½×E for B, and 2×E for F in eq.2: A = 1½×E + 2×E which makes A = 3½×E
Hint #6
Substitute 4×E for D, 3½×E for A, and 1½×E for B in eq.6: 4×E + E = 3½×E + 1½×E + C which becomes 5×E = 5×E + C Subtract 5×E from both sides of the equation above: 5×E – 5×E = 5×E + C – 5×E which makes 0 = C
Solution
Substitute 3½×E for A, 1½×E for B, 0 for C, 4×E for D, and 2×E for F in eq.1: 3½×E + 1½×E + 0 + 4×E + E + 2×E = 24 which simplifies to 12×E = 24 Divide both sides of the above equation by 12: 12×E ÷ 12 = 24 ÷ 12 which means E = 2 making A = 3½×E = 3½ × 2 = 7 B = 1½×E = 1½ × 2 = 3 D = 4×E = 4 × 2 = 8 F = 2×E = 2 × 2 = 4 and ABCDEF = 730824