Puzzle for August 24, 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
Help Area
Hint #1
In eq.6, replace F with A + C (from eq.2): E + A + C = A + B + D Subtract A from each side of the equation above: E + A + C - A = A + B + D - A which becomes E + C = B + D which may be written as eq.6a) C + E = B + D
Hint #2
In eq.5, replace C + E with B + D (from eq.6a): B + D = A + D Subtract D from each side of the above equation: B + D - D = A + D - D which makes B = A
Hint #3
In eq.3, substitute A for B: eq.3a) C = A + E
Hint #4
Substitute A + E for C (from eq.3a) in eq.2: F = A + A + E which becomes eq.2a) F = 2×A + E
Hint #5
Substitute A for B, and 2×A + E for F (from eq.2a) in eq.4: D = A + A + 2×A + E which becomes eq.4a) D = 4×A + E
Hint #6
Substitute A + E for C (from eq.3a), A for B, and 4×A + E for D (from eq.4a) in eq.6a: A + E + E = A + 4×A + E which becomes A + 2×E = 5×A + E Subtract A and E from both sides of the above equation: A + 2×E - A - E = 5×A + E - A - E which makes E = 4×A
Hint #7
Substitute 4×A for E in eq.3a: C = A + 4×A which makes C = 5×A
Hint #8
Substitute 4×A for E in eq.4a: D = 4×A + 4×A which makes D = 8×A
Hint #9
Substitute 4×A for E in eq.2a: F = 2×A + 4×A which makes F = 6×A
Solution
Substitute A for B, 5×A for C, 8×A for D, 4×A for E, and 6×A for F in eq.1: A + A + 5×A + 8×A + 4×A + 6×A = 25 which simplifies to 25×A = 25 Divide both sides of the equation above by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = A = 1 C = 5×A = 5 × 1 = 5 D = 8×A = 8 × 1 = 8 E = 4×A = 4 × 1 = 4 F = 6×A = 6 × 1 = 6 and ABCDEF = 115846