Puzzle for August 27, 2020 ( )
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 C to both sides of eq.4: D – C + C = B + F + C which becomes D = B + F + C In eq.2, replace D with B + F + C: A + B + C = B + F + C Subtract B and C from each side of the above equation: A + B + C – B – C = B + C + F – B – C which simplifies to A = F
Hint #2
In eq.5, replace F with A: A + A = B + C + D which becomes eq.5a) 2×A = B + C + D
Hint #3
Add D to both sides of eq.2: A + B + C + D = D + D which becomes A + B + C + D = 2×D In the equation above, substitute 2×A for B + C + D (from eq.5a): A + 2×A = 2×D which makes 3×A = 2×D Divide both sides by 2: 3×A ÷ 2 = 2×D ÷ 2 which makes eq.2a) 1½×A = D
Hint #4
Substitute 1½×A for D, and A for F in eq.6: B – C + 1½×A = A – B + A which becomes B – C + 1½×A = 2×A – B In the above equation, add C and B to both sides, and subtract 2×A from both sides: B – C + 1½×A + C + B – 2×A = 2×A – B + C + B – 2×A which becomes eq.6a) 2×B – ½×A = C
Hint #5
Substitute 2×B – ½×A for C (from eq.6a), and 1½×A for D in eq.5a: 2×A = B + 2×B – ½×A + 1½×A which becomes 2×A = 3×B + A Subtract A from each side of the equation above: 2×A – A = 3×B + A – A which makes A = 3×B and also makes F = A = 3×B
Hint #6
Substitute (3×B) for A in eq.2a: 1½×(3×B) = D which makes 4½×B = D
Hint #7
Substitute 3×B for A in eq.6a: 2×B – ½×(3×B) = C which becomes 2×B – 1½×B = C which makes ½×B = C
Hint #8
Substitute ½×B for C, and 3×B for F in eq.3: E – ½×B = 3×B Add ½×B to each side of the above equation: E – ½×B + ½×B = 3×B + ½×B which makes E = 3½×B
Solution
Substitute 3×B for A and F, ½×B for C, 4½×B for D, and 3½×B for E in eq.1: 3×B + B + ½×B + 4½×B + 3½×B + 3×B = 31 which simplifies to 15½×B = 31 Divide both sides of the above equation by 15½: 31×B ÷ 15½ = 31 ÷ 15½ which means B = 2 making A = F = 3×B = 3 × 2 = 6 C = ½×B = ½ × 2 = 1 D = 4½×B = 4½ × 2 = 9 E = 3½×B = 3½ × 2 = 7 and ABCDEF = 621976