Puzzle for November 15, 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.6, replace A - E with D - A (from eq.5): eq.6a) E - F = D - A
Hint #2
In eq.6a, replace D with E + F (from eq.4): E - F = E + F - A Subtract E from each side of the equation above: E - F - E = E + F - A - E which becomes -F = F - A Add F and A to both sides: -F + F + A = F - A + F + A which makes A = 2×F
Hint #3
In eq.6, substitute 2×F for A: E - F = 2×F - E Add F and E to both sides of the above equation: E - F + F + E = 2×F - E + F + E which makes 2×E = 3×F Divide both sides by 2: 2×E ÷ 2 = 3×F ÷ 2 which makes E = 1½×F
Hint #4
Substitute 1½×F for E in eq.4: D = 1½×F + F which makes D = 2½×F
Hint #5
Substitute 2×F for A, and 2½×F for D in eq.3: C = 2×F + 2½×F which makes C = 4½×F
Hint #6
Substitute 2½×F for D in eq.2: B = 2½×F + F which makes B = 3½×F
Solution
Substitute 2×F for A, 3½×F for B, 4½×F for C, 2½×F for D, and 1½×F for E in eq.1: 2×F + 3½×F + 4½×F + 2½×F + 1½×F + F = 30 which simplifies to 15×F = 30 Divide both sides of the above equation by 15: 15×F ÷ 15 = 30 ÷ 15 which means F = 2 making A = 2×F = 2 × 2 = 4 B = 3½×F = 3½ × 2 = 7 C = 4½×F = 4½ × 2 = 9 D = 2½×F = 2½ × 2 = 5 E = 1½×F = 1½ × 2 = 3 and ABCDEF = 479532