Puzzle for November 9, 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
Subtract A from each side of eq.2: C - A = A + E + F - A which becomes eq.2a) C - A = E + F
Hint #2
In eq.4, replace E + F with C - A (from eq.2a): C - A = A - (C - A) which becomes C - A = A - C + A which becomes C - A = 2×A - C Add A and C to both sides of the above equation: C - A + A + C = 2×A - C + A + C which makes 2×C = 3×A Divide both sides by 2: 2×C ÷ 2 = 3×A ÷ 2 which makes eq.4a) C = 1½×A
Hint #3
Add E and B to both sides of eq.3: F - E + E + B = E - B + E + B which becomes eq.3a) F + B = 2×E
Hint #4
In eq.5, replace C with A + E + F (from eq.2): A + D + F = A + E + F + E - B which becomes A + D + F = A + 2×E + F - B In the above equation, subtract A and F from both sides, and add B to both sides: A + D + F - A - F + B = A + 2×E + F - B - A - F + B which simplifies to eq.5a) D + B = 2×E
Hint #5
In eq.5a, substitute F + B for 2×E (from eq.3a): D + B = F + B Subtract B from both sides of the above equation: D + B - B = F + B - B which makes D = F
Hint #6
In eq.6, add E and A to both sides, and subtract B and D from both sides: A + B - E + E + A - B - D = C + D - A + E + A - B - D which becomes eq.6a) 2×A - D = C + E - B
Hint #7
In eq.5, substitute D for F, and 2×A - D for C + E - B (from eq.6a): A + D + D = 2×A - D which becomes A + 2×D = 2×A - D In the above equation, subtract A from both sides, and add D to both sides: A + 2×D - A + D = 2×A - D - A + D which makes 3×D = A
Hint #8
Substitute (3×D) for A in eq.4a: C = 1½×(3×D) which makes C = 4½×D
Hint #9
Substitute 4½×D for C, 3×D for A, and D for F in eq.2: 4½×D = 3×D + E + D which becomes 4½×D = 4×D + E Subtract 4×D from each side of the above equation: 4½×D - 4×D = 4×D + E - 4×D which makes ½×D = E
Hint #10
Substitute (½×D) for E in eq.5a: D + B = 2×(½×D) which becomes D + B = D Subtract D from each side of the equation above: D + B - D = D - D which makes B = 0
Solution
Substitute 3×D for A, 0 for B, 4½×D for C, ½×D for E, and D for F in eq.1: 3×D + 0 + 4½×D + D + ½×D + D = 20 which simplifies to 10×D = 20 Divide both sides of the above equation by 10: 10×D ÷ 10 = 20 ÷ 10 which means D = 2 making A = 3×D = 3 × 2 = 6 C = 4½×D = 4½ × 2 = 9 E = ½×D = ½ × 2 = 1 F = D = 2 and ABCDEF = 609212