Puzzle for October 12, 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 non-negative integer.
Scratchpad
Help Area
Hint #1
Add D to both sides of eq.3: C - D + D = D + F + D which becomes eq.3a) C = 2×D + F
Hint #2
In eq.6, replace C with 2×D + F (from eq.3a): A + 2×D + F = D + E + F Subtract D and F from each side of the above equation: A + 2×D + F - D - F = D + E + F - D - F which becomes eq.6a) A + D = E
Hint #3
In eq.2, replace E with A + D (from eq.6a): D + A + D = B - D which becomes 2×D + A = B - D Add D to both sides of the equation above: 2×D + A + D = B - D + D which becomes eq.2a) 3×D + A = B
Hint #4
In eq.4, substitute 3×D + A for B (from eq.2a): 3×D + A + D = A + F which becomes 4×D + A = A + F Subtract A from each side of the equation above: 4×D + A - A = A + F - A which makes 4×D = F
Hint #5
Substitute 4×D for F in eq.5: D + 4×D = B - 4×D which becomes 5×D = B - 4×D Add 4×D to both sides of the above equation: 5×D + 4×D = B - 4×D + 4×D which makes 9×D = B
Hint #6
Substitute 4×D for F in eq.3: C - D = D + 4×D which becomes C - D = 5×D Add D to both sides of the equation above: C - D + D = 5×D + D which makes C = 6×D
Hint #7
Substitute 9×D for B in eq.2: D + E = 9×D - D which becomes D + E = 8×D Subtract D from each side of the above equation: D + E - D = 8×D - D which makes E = 7×D
Hint #8
Substitute 9×D for B, and 4×D for F in eq.4: 9×D + D = A + 4×D which becomes 10×D = A + 4×D Subtract 4×D from both sides of the above equation: 10×D - 4×D = A + 4×D - 4×D which makes 6×D = A
Solution
Substitute 6×D for A and C, 9×D for B, 7×D for E, and 4×D for F in eq.1: 6×D + 9×D + 6×D + D + 7×D + 4×D = 33 which simplifies to 33×D = 33 Divide both sides of the above equation by 33: 33×D ÷ 33 = 33 ÷ 33 which means D = 1 making A = C = 6×D = 6 × 1 = 6 B = 9×D = 9 × 1 = 9 E = 7×D = 7 × 1 = 7 F = 4×D = 4 × 1 = 4 and ABCDEF = 696174