Puzzle for May 15, 2021 ( )
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 D + F with A + B - D (from eq.6): A + E = B + A + B - D which becomes A + E = A + 2×B - D In the above equation, subtract A from both sides, and add D to both sides: A + E - A + D = A + 2×B - D - A + D which becomes E + D = 2×B which is the same as eq.4a) D + E = 2×B
Hint #2
In eq.3, replace D + E with 2×B (from eq.4a): 2×B = B + C Subtract B from each side of the equation above: 2×B - B = B + C - B which makes B = C
Hint #3
In eq.2, substitute B for C: B + E = A + B Subtract B from both sides of the above equation: B + E - B = A + B - B which makes E = A
Hint #4
Substitute B for C in eq.5: B + B + D = A - B + F which becomes 2×B + D = A - B + F In the above equation, subtract A from both sides, and add B to both sides: 2×B + D - A + B = A - B + F - A + B which becomes eq.5a) 3×B + D - A = F
Hint #5
Substitute 3×B + D - A for F (from eq.5a) in eq.6: D + 3×B + D - A = A + B - D which becomes 3×B + 2×D - A = A + B - D In the equation above, add A to both sides, and subtract B and 2×D from both sides: 3×B + 2×D - A + A - B - 2×D = A + B - D + A - B - 2×D which simplifies to eq.6a) 2×B = 2×A - 3×D
Hint #6
Substitute A for E, and 3×B + D - A for F (from eq.5a) in eq.4: A + A = B + D + 3×B + D - A which becomes 2×A = 4×B + 2×D - A Add A to each side of the equation above: 2×A + A = 4×B + 2×D - A + A which becomes 3×A = 4×B + 2×D which may be written as eq.4b) 3×A = 2×(2×B) + 2×D
Hint #7
Substitute 2×A - 3×D for 2×B (from eq.6a) in eq.4b: 3×A = 2×(2×A - 3×D) + 2×D which becomes 3×A = 4×A - 6×D + 2×D which becomes 3×A = 4×A - 4×D In the above equation, add 4×D to each side, and subtract 3×A from each side: 3×A + 4×D - 3×A = 4×A - 4×D + 4×D - 3×A which makes 4×D = A and also makes E = A = 4×D
Hint #8
Substitute (4×D) for A in eq.6a: 2×B = 2×(4×D) - 3×D which becomes 2×B = 8×D - 3×D which makes 2×B = 5×D Divide both sides of the above equation by 2: 2×B ÷ 2 = 5×D ÷ 2 which makes B = 2½×D and also makes C = B = 2½×D
Hint #9
Substitute (2½×D) for B, and 4×D for A in eq.5a: 3×(2½×D) + D - 4×D = F which becomes 7½×D - 3×D = F which makes 4½×D = F
Solution
Substitute 4×D for A and E, 2½×D for B and C, and 4½×D for F in eq.1: 4×D + 2½×D + 2½×D + D + 4×D + 4½×D = 37 which simplifies to 18½×D = 37 Divide both sides of the equation above by 18½: 18½×D ÷ 18½ = 37 ÷ 18½ which means D = 2 making A = E = 4×D = 4 × 2 = 8 B = C = 2½×D = 2½ × 2 = 5 F = 4½×D = 4½ × 2 = 9 and ABCDEF = 855289