Puzzle for July 4, 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.4, replace D with A + C (from eq.3): C + E = A + A + C which becomes C + E = 2×A + C Subtract C from each side of the equation above: C + E – C = 2×A + C – C which makes E = 2×A
Hint #2
In eq.5, replace E with 2×A: 2×A – B = B – A Add B and A to both sides of the above equation: 2×A – B + B + A = B – A + B + A which makes 3×A = 2×B Divide both sides by 2: 3×A ÷ 2 = 2×B ÷ 2 which makes 1½×A = B
Hint #3
In eq.2, substitute 1½×A for B, and 2×A for E: C = 1½×A + 2×A which makes C = 3½×A
Hint #4
Substitute 3½×A for C in eq.3: D = A + 3½×A which makes D = 4½×A
Hint #5
Substitute 1½×A for B, and 3½×A for C in eq.6: 1½×A + 3½×A = A + F which becomes 5×A = A + F Subtract A from each side of the above equation: 5×A – A = A + F – A which makes 4×A = F
Solution
Substitute 1½×A for B, 3½×A for C, 4½×A for D, 2×A for E, and 4×A for F in eq.1: A + 1½×A + 3½×A + 4½×A + 2×A + 4×A = 33 which simplifies to 16½×A = 33 Divide both sides of the above equation by 16½: 16½×A ÷ 16½ = 33 ÷ 16½ which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = 3½×A = 3½ × 2 = 7 D = 4½×A = 4½ × 2 = 9 E = 2×A = 2 × 2 = 4 F = 4×A = 4 × 2 = 8 and ABCDEF = 237948