Puzzle for April 7, 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.3, replace F with B + C (from eq.2): B + B + C = C + E which becomes 2×B + C = C + E Subtract C from each side of the equation above: 2×B + C - C = C + E - C which makes eq.3a) 2×B = E
Hint #2
In eq.4, replace E with 2×B: 2×B = A + B + D Subtract B from each side of the above equation: 2×B - B = A + B + D - B which becomes eq.4a) B = A + D
Hint #3
In eq.5, replace E with 2×B: B + 2×B - A = C + F - B which becomes 3×B - A = C + F - B Add A and B to both sides of the equation above: 3×B - A + A + B = C + F - B + A + B which becomes 4×B = C + F + A which may be written as eq.5a) 4×B = A + C + F
Hint #4
In eq.6, substitute B for A + D (from eq.4a), and 2×B for E: C + F - A = B + 2×B which becomes C + F - A = 3×B Add A to both sides of the above equation: C + F - A + A = 3×B + A which becomes eq.6a) C + F = 3×B + A
Hint #5
Substitute 3×B + A for C + F (from eq.6a) into eq.5a: 4×B = A + 3×B + A which becomes 4×B = 2×A + 3×B Subtract 3×B from each side of the equation above: 4×B - 3×B = 2×A + 3×B - 3×B which makes B = 2×A
Hint #6
Substitute (2×A) for B in eq.3a: 2×(2×A) = E which makes 4×A = E
Hint #7
Substitute 2×A for B in eq.4a: 2×A = A + D Subtract A from each side of the above equation: 2×A - A = A + D - A which makes A = D
Hint #8
Substitute 2×A for B in eq.2: eq.2a) F = 2×A + C
Hint #9
Substitute 2×A + C for F (from eq.2a), and (2×A) for B in eq.6a: C + 2×A + C = 3×(2×A) + A which becomes 2×C + 2×A = 6×A + A which becomes 2×C + 2×A = 7×A Subtract 2×A from each side of the equation above: 2×C + 2×A - 2×A = 7×A - 2×A which makes 2×C = 5×A Divide both sides by 2: 2×C ÷ 2 = 5×A ÷ 2 which makes C = 2½×A
Hint #10
Substitute 2½×A for C in eq.2a: F = 2×A + 2½×A which makes F = 4½×A
Solution
Substitute 2×A for B, 2½×A for C, A for D, 4×A for E, and 4½×A for F in eq.1: A + 2×A + 2½×A + A + 4×A + 4½×A = 30 which simplifies to 15×A = 30 Divide both sides of the above equation by 15: 15×A ÷ 15 = 30 ÷ 15 which means A = 2 making B = 2×A = 2 × 2 = 4 C = 2½×A = 2½ × 2 = 5 D = A = 2 E = 4×A = 4 × 2 = 8 F = 4½×A = 4½ × 2 = 9 and ABCDEF = 245289