Puzzle for July 13, 2021 ( )
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
Add B and C to both sides of eq.4: C – B + B + C = A – C + B + C which becomes eq.4a) 2×C = A + B In eq.2, replace A + B with 2×C (from eq.4a): E = 2×C
Hint #2
Add C to both sides of eq.5: B + C + C = F – C + C which becomes eq.5a) B + 2×C = F In eq.5a, replace F with B + D (from eq.3): B + 2×C = B + D Subtract B from both sides of the above equation: B + 2×C – B = B + D – B which makes 2×C = D
Hint #3
In eq.6, replace E and D with 2×C: 2×C + F – A = A + 2×C In the above equation, subtract 2×C from both sides, and add A to both sides: 2×C + F – A – 2×C + A = A + 2×C – 2×C + A which makes eq.6a) F = 2×A
Hint #4
In eq.5a, substitute A + B for 2×C (from eq.4a), and 2×A for F (from eq.6a): B + A + B = 2×A which becomes 2×B + A = 2×A Subtract A from both sides of the equation above: 2×B + A – A = 2×A – A which makes 2×B = A
Hint #5
Substitute 2×B for A in eq.2: E = 2×B + B which makes E = 3×B and also means D = E = 3×B
Hint #6
Substitute 2×B for A in eq.4a: 2×C = 2×B + B which becomes 2×C = 3×B Divide both sides of the above equation by 2: 2×C ÷ 2 = 3×B ÷ 2 which makes C = 1½×B
Hint #7
Substitute (2×B) for A in eq.6a: F = 2×(2×B) which makes F = 4×B
Solution
Substitute 2×B for A, 1½×B for C, 3×B for D and E, 4×B for F in eq.1: 2×B + B + 1½×B + 3×B + 3×B + 4×B = 29 which simplifies to 14½×B = 29 Divide both sides of the above equation by 14½: 14½×B ÷ 14½ = 29 ÷ 14½ which means B = 2 making A = 2×B = 2 × 2 = 4 C = 1½×B = 1½ × 2 = 3 D = E = 3×B = 3 × 2 = 6 F = 4×B = 4 × 2 = 8 and ABCDEF = 423668