Puzzle for October 8, 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
eq.6 may be written as: C + D = (A + B + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × (C + D) = 4 × (A + B + E + F) ÷ 4 which becomes eq.6a) 4×C + 4×D = A + B + E + F
Hint #2
Add C to both sides of eq.4: C + F + C = A + B - C + C which becomes eq.4a) 2×C + F = A + B
Hint #3
In eq.6a, replace A + B with 2×C + F (from eq.4a): 4×C + 4×D = 2×C + F + E + F which becomes 4×C + 4×D = 2×C + 2×F + E Subtract 2×C from each side of the equation above: 4×C + 4×D - 2×C = 2×C + 2×F + E - 2×C which becomes eq.6b) 2×C + 4×D = 2×F + E
Hint #4
eq.6a may be written as: 4×C + 4×D = A + F + B + E In the above equation, replace A + F with B + E (from eq.3): 4×C + 4×D = B + E + B + E which becomes 4×C + 4×D = 2×B + 2×E Divide both sides by 2: (4×C + 4×D) ÷ 2 = (2×B + 2×E) ÷ 2 which becomes eq.6c) 2×C + 2×D = B + E
Hint #5
Subtract the left and right sides of eq.6c from the left and right sides of eq.6b, respectively: 2×C + 4×D - (2×C + 2×D) = 2×F + E - (B + E) which becomes 2×C + 4×D - 2×C - 2×D = 2×F + E - B - E which becomes 2×D = 2×F - B Add B to both sides of the above equation: 2×D + B = 2×F - B + B which becomes eq.6d) 2×D + B = 2×F
Hint #6
Add D to both sides of eq.5: B + D + D = E + F - D + D which becomes B + 2×D = E + F which may be written as eq.5a) 2×D + B = E + F
Hint #7
In eq.5a, substitute 2×F for 2×D + B (from eq.6d): 2×F = E + F Subtract F from each side of the equation above: 2×F - F = E + F - F which makes F = E
Hint #8
Substitute E for F in eq.3: B + E = A + E Subtract E from each side of the above equation: B + E - E = A + E - E which makes B = A
Hint #9
Substitute B for A in eq.2: D = B - E Add E to both sides of the above equation: D + E = B - E + E which becomes eq.2a) D + E = B
Hint #10
Substitute D + E for B (from eq.2a), and E for F in eq.6d: 2×D + D + E = 2×E which becomes 3×D + E = 2×E Subtract E from both sides of the above equation: 3×D + E - E = 2×E - E which makes 3×D = E and also makes 3×D = E = F
Hint #11
Substitute 3×D for E in eq.2a: D + 3×D = B which makes 4×D = B and also makes 4×D = B = A
Hint #12
Substitute (3×D) for F and E in eq.6b: 2×C + 4×D = 2×(3×D) + (3×D) which becomes 2×C + 4×D = 6×D + 3×D which becomes 2×C + 4×D = 9×D Subtract 4×D from both sides of the above equation: 2×C + 4×D - 4×D = 9×D - 4×D which makes 2×C = 5×D Divide both sides by 2: 2×C ÷ 2 = 5×D ÷ 2 which makes C = 2½×D
Solution
Substitute 4×D for A and B, 2½×D for C, and 3×D for E for F in eq.1: 4×D + 4×D + 2½×D + D + 3×D + 3×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 = B = 4×D = 4 × 2 = 8 C = 2½×D = 2½ × 2 = 5 E = F = 3×D = 3 × 2 = 6 and ABCDEF = 885266