Puzzle for August 10, 2023 ( )
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.5, replace D with C + E (from eq.2), and E + F with B + C (from eq.4): B + C + E = A + B + C Subtract B and C from each side of the equation above: B + C + E - B - C = A + B + C - B - C which simplifies to E = A
Hint #2
In eq.3, replace D with C + E (from eq.2): B + F = C + C + E which becomes eq.3a) B + F = 2×C + E Subtract B and E from each side of eq.3a above: B + F - B - E = 2×C + E - B - E which becomes eq.3b) F - E = 2×C - B
Hint #3
Add the left and right sides of eq.3b to the left and right sides of eq.4, respectively: E + F + F - E = B + C + 2×C - B which becomes 2×F = 3×C Divide both sides of the above equation by 2: 2×F ÷ 2 = 3×C ÷ 2 which makes F = 1½×C
Hint #4
In eq.6, substitute E for A, C + E for D (from eq.2), and 1½×C for F: E + B = C + C + E + 1½×C which becomes E + B = 3½×C + E Subtract E from each side of the equation above: E + B - E = 3½×C + E - E which makes B = 3½×C
Hint #5
Substitute 3½×C for B, and 1½×C for F in eq.3a: 3½×C + 1½×C = 2×C + E which becomes 5×C = 2×C + E Subtract 2×C from each side of the above equation: 5×C - 2×C = 2×C + E - 2×C which makes 3×C = E and also makes 3×C = E = A
Hint #6
Substitute 3×C for E in eq.2: C + 3×C = D which makes 4×C = D
Solution
Substitute 3×C for A and E, 3½×C for B, 4×C for D, and 1½×C for F in eq.1: 3×C + 3½×C + C + 4×C + 3×C + 1½×C = 32 which simplifies to 16×C = 32 Divide both sides of the above equation by 16: 16×C ÷ 16 = 32 ÷ 16 which means C = 2 making A = E = 3×C = 3 × 2 = 6 B = 3½×C = 3½ × 2 = 7 D = 4×C = 4 × 2 = 8 F = 1½×C = 1½ × 2 = 3 and ABCDEF = 672863