Puzzle for January 19, 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 positive integer.
Scratchpad
Help Area
Hint #1
In eq.4, replace E with C + F (from eq.3): B + C = A + C + F Subtract C from each side of the equation above: B + C - C = A + C + F - C which becomes eq.4a) B = A + F
Hint #2
In eq.2, replace A + F with B (from eq.4a): D = B
Hint #3
In eq.5, replace B with A + F (from eq.4a): F - A = A + F - F which becomes F - A = A Add A to both sides of the above equation: F - A + A = A + A which makes F = 2×A
Hint #4
In eq.4a, substitute 2×A for F: B = A + 2×A which makes B = 3×A and also makes D = B = 3×A
Hint #5
In eq.3, substitute 2×A for F: eq.3a) E = C + 2×A
Hint #6
In eq.6, substitute (3×A) for D, (C + 2×A) for E (from eq.3a), and 2×A for F: C × (3×A) = (A × (C + 2×A)) + 2×A which becomes C × (3×A) = (A × C) + A×2×A + 2×A which may be written as eq.6a) 3×(A × C) = (A × C) + 2×A×(A + 1)
Hint #7
Subtract (A × C) from each side of eq.6a: 3×(A × C) - (A × C) = (A × C) + 2×A×(A + 1) - (A × C) which becomes 2×(A × C) = 2×A×(A + 1) which may be written as 2×A×C = 2×A×(A + 1) Divide both sides by 2×A: 2×A×C ÷ 2×A = 2×A×(A + 1) ÷ 2×A which makes eq.6b) C = A + 1
Hint #8
Substitute A + 1 for C (from eq.6b) in eq.3a: E = A + 1 + 2×A which becomes eq.3b) E = 3×A + 1
Hint #9
Substitute 3×A for B and D, A + 1 for C (from eq.6b), 3×A + 1 for E (from eq.3b), and 2×A for F in eq.1: A + 3×A + A + 1 + 3×A + 3×A + 1 + 2×A = 28 which simplifies to 13×A + 2 = 28 Subtract 2 from each side of the above equation: 13×A + 2 - 2 = 28 - 2 which makes 13×A = 26 Divide both sides by 13: 13×A ÷ 13 = 26 ÷ 13 which means A = 2
Solution
Since A = 2, then: B = D = 3×A = 3 × 2 = 6 C = A + 1 = 2 + 1 = 3 (from eq.6b) E = 3×A + 1 = 3×2 + 1 = 6 + 1 = 7 (from eq.3b) F = 2×A = 2 × 2 = 4 and ABCDEF = 263674