Puzzle for September 6, 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
In eq.5, replace B + C + F with D (from eq.6): eq.5a) A + D = E
Hint #2
In eq.5a, replace E with C + D (from eq.4): A + D = C + D Subtract D from each side of the above equation: A + D - D = C + D - D which makes A = C
Hint #3
In eq.2, substitute A for C: A + A = F which makes 2×A = F
Hint #4
Substitute 2×A for F in eq.3: 2×A - B = A + B In the equation above, add B to both sides, and subtract A from both sides: 2×A - B + B - A = A + B + B - A which becomes A = 2×B Divide both sides by 2: A ÷ 2 = 2×B ÷ 2 which makes ½×A = B
Hint #5
Substitute ½×A for B, A for C, and 2×A for F in eq.6: ½×A + A + 2×A = D which makes 3½×A = D
Hint #6
Substitute A for C, and 3½×A for D in eq.4: A + 3½×A = E which makes 4½×A = E
Solution
Substitute ½×A for B, A for C, 3½×A for D, 4½×A for E, and 2×A for F in eq.1: A + ½×A + A + 3½×A + 4½×A + 2×A = 25 which simplifies to 12½×A = 25 Divide both sides of the above equation by 12½: 12½×A ÷ 12½ = 25 ÷ 12½ which means A = 2 making B = ½×A = ½ × 2 = 1 C = A = 2 D = 3½×A = 3½ × 2 = 7 E = 4½×A = 4½ × 2 = 9 F = 2×A = 2 × 2 = 4 and ABCDEF = 212794