Puzzle for October 26, 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 positive integer.
* CD and BC are 2-digit numbers (not C×D or B×C).
Scratchpad
Help Area
Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: E + F - (D + E) = B + D - (B + F) which becomes E + F - D - E = B + D - B - F which becomes F - D = D - F Add D and F to both sides of the above equation: F - D + D + F = D - F + D + F which becomes 2×F = 2×D Divide both sides by 2: 2×F ÷ 2 = 2×D ÷ 2 which makes F = D
Hint #2
In eq.2, replace F with D: E + D = B + D Subtract D from each side of the equation above: E + D - D = B + D - D which makes E = B
Hint #3
eq.4 may be written as: F = (B + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (B + D + E) ÷ 3 which becomes eq.4a) 3×F = B + D + E
Hint #4
In eq.4a, replace F with D, and E with B: 3×D = B + D + B which becomes 3×D = 2×B + D Subtract D from each side of the above equation: 3×D - D = 2×B + D - D which makes 2×D = 2×B Divide both sides by 2: 2×D ÷ 2 = 2×B ÷ 2 which makes D = B and also makes F = D = B = E
Hint #5
eq.5 may be written as: 10×C + D - (10×B + C) = B + E + F which becomes 10×C + D - 10×B - C = B + E + F which becomes eq.5a) 9×C + D - 10×B = B + E + F
Hint #6
Substitute B + D for E + F (from eq.2) in eq.5a: 9×C + D - 10×B = B + B + D which becomes 9×C + D - 10×B = 2×B + D In the above equation, subtract D from both sides, and add 10×B to both sides: 9×C + D - 10×B - D + 10×B = 2×B + D - D + 10×B which makes 9×C = 12×B Divide both sides by 12: 9×C ÷ 12 = 12×B ÷ 12 which makes ¾×C = B and also makes ¾×C = F = D = B = E
Hint #7
Substitute ¾×C for B, F, D, and E in eq.6: A × C = (¾×C × ¾×C) + (¾×C × ¾×C) which becomes A × C = 2×((¾×C)²) which becomes A × C = 1⅛×C² Divide both sides of the above equation by C: (A × C) ÷ C = 1⅛×C² ÷ C which makes A = 1⅛×C
Solution
Substitute 1⅛×C for A, and ¾×C for B, D, E, and F in eq.1: 1⅛×C + ¾×C + C + ¾×C + ¾×C + ¾×C = 41 which simplifies to 5⅛×C = 41 Divide both sides of the above equation by 5⅛: 5⅛×C ÷ 5⅛ = 41 ÷ 5⅛ which makes C = 8 making A = 1⅛×C = 1⅛ × 8 = 9 B = D = E = F = ¾×C = ¾ × 8 = 6 and ABCDEF = 968666