Puzzle for May 17, 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.4, replace B with D + E (from eq.2): D + E + D = A + E which becomes 2×D + E = A + E Subtract E from each side of the equation above: 2×D + E - E = A + E - E which makes 2×D = A
Hint #2
Add E to both sides of eq.5: A + D + E = C - E + F + E which becomes A + D + E = C + F In the above equation, replace D + E with B (from eq.2): eq.5a) A + B = C + F
Hint #3
Subtract the left and right sides of eq.5a from the left and right sides of eq.3, respectively: A + C - (A + B) = B + F - (C + F) which becomes A + C - A - B = B + F - C - F which becomes C - B = B - C Add B and C to both sides of the above equation: C - B + B + C = B - C + B + C which makes 2×C = 2×B Divide both sides by 2: 2×C ÷ 2 = 2×B ÷ 2 which makes C = B
Hint #4
In eq.5a, substitute 2×D for A, and B for C: 2×D + B = B + F Subtract B from each side of the above equation: 2×D + B - B = B + F - B which makes 2×D = F
Hint #5
In eq.6, substitute 2×D for A and F, and B for C: 2×D + D + 2×D = B + B - 2×D - 2×D which becomes 5×D = 2×B - 4×D Add 4×D to both sides of the above equation: 5×D + 4×D = 2×B - 4×D + 4×D which makes 9×D = 2×B Divide both sides by 2: 9×D ÷ 2 = 2×B ÷ 2 which makes 4½×D = B and also makes 4½×D = B = C
Hint #6
Substitute 4½×D for B in eq.2: 4½×D = D + E Subtract D from each side of the above equation: 4½×D - D = D + E - D which makes 3½×D = E
Solution
Substitute 2×D for A and F, 4½×D for B and C, and 3½×D for E in eq.1: 2×D + 4½×D + 4½×D + D + 3½×D + 2×D = 35 which simplifies to 17½×D = 35 Divide both sides of the above equation by 17½: 17½×D ÷ 17½ = 35 ÷ 17½ which means D = 2 making A = F = 2×D = 2 × 2 = 4 B = C = 4½×D = 4½ × 2 = 9 E = 3½×D = 3½ × 2 = 7 and ABCDEF = 499274