Puzzle for March 9, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
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Hint #1
eq.6 may be written as: F = (A + B + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (A + B + D) ÷ 3 which becomes eq.6a) 3×F = A + B + D which may be written as eq.6b) 3×F = A + D + B
Hint #2
In eq.6b, replace A + D with B (from eq.1): 3×F = B + B which becomes 3×F = 2×B Divide both sides of the above equation by 2: 3×F ÷ 2 = 2×B ÷ 2 which makes 1½×F = B
Hint #3
In eq.2, replace B with 1½×F: E + F = 1½×F Subtract F from each side of the equation above: E + F – F = 1½×F – F which makes E = ½×F
Hint #4
In eq.3, substitute ½×F for E, and 3×F for A + B + D (from eq.6a): C × ½×F = 3×F Divide both sides of the above equation by ½×F: C × ½×F ÷ ½×F = 3×F ÷ ½×F which makes C = 6
Hint #5
Substitute ½×F for E, 1½×F for B, and 6 for C in eq.4: (A × F) + ½×F = (1½×F × 6) – ½×F which becomes (A × F) + ½×F = 9×F – ½×F which becomes (A × F) + ½×F = 8½×F Subtract ½×F from both sides of the above equation: (A × F) + ½×F – ½×F = 8½×F – ½×F which becomes A × F = 8×F Divide both sides by F: A × F ÷ F = 8×F ÷ F which makes A = 8
Hint #6
Substitute 1½×F for B, and 8 for A in eq.1: 1½×F = 8 + D Subtract 8 from each side of the above equation: 1½×F – 8 = 8 + D – 8 which becomes eq.1a) 1½×F – 8 = D
Solution
Substitute 8 for A, ½×F for E, and 6 for C in eq.5: (8 × ½×F) + F = (6 × F) – 6 which becomes 4×F + F = 6×F – 6 which becomes 5×F = 6×F – 6 In the equation above, subtract 5×F from both sides, and add 6 to both sides: 5×F – 5×F + 6 = 6×F – 6 – 5×F + 6 which makes 6 = F making B = 1½×F = 1½ × 6 = 9 D = 1½×F – 8 = 1½×6 – 8 = 9 – 8 = 1 (from eq.1a) E = ½×F = ½ × 6 = 3 and ABCDEF = 896136