Puzzle for August 3, 2019 ( )
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.6, solve for C – D by subtracting D from each side and adding B to each side: C – B – D + B = B + D – F – D + B which becomes C – D = 2×B – F In eq.2, replace C – D with 2×B – F: 2×B – F = A – F Add F to both sides of the above equation: 2×B – F + F = A – F + F which makes eq.2a) 2×B = A
Hint #2
In eq.4, replace A with 2×B: B + C = 2×B + F Subtract B from both sides of the above equation: B + C – B = 2×B + F – B which becomes eq.4a) C = B + F
Hint #3
In eq.3, substitute (B + F) for C (from eq.4a): E – F = B – (B + F) which is equivalent to E – F = B – B – F which becomes E – F = –F Add F to both sides of the above equation: E – F + F = –F + F which makes E = 0
Hint #4
In eq.6, substitute (B + F) for C (from eq.4a): B + F – B = B + D – F which becomes F = B + D – F In the above equation, solve for D by adding F to both sides and subtracting B from both sides: F + F – B = B + D – F + F – B which simplifies to eq.6a) 2×F – B = D
Hint #5
Substitute 2×B for A, 0 for E, and B + F for C (from eq.4a) in eq.5: 2×B + B + 0 = B + F + D – 2×B which becomes eq.5a) 3×B = F + D – B
Hint #6
Substitute 2×F – B for D (from eq.6a) in eq.5a: 3×B = F + 2×F – B – B which becomes 3×B = 3×F – 2×B Add 2×B to both sides of the above equation: 3×B + 2×B = 3×F – 2×B + 2×B which means eq.5b) 5×B = 3×F
Hint #7
Multiply both sides of eq.4a by 5: 5×C = 5×(B + F) which becomes 5×C = 5×B + 5×F Substitute 3×F for 5×B (from eq.5b) in the above equation: 5×C = 3×F + 5×F which means 5×C = 8×F Divide both sides by 8: 5×C ÷ 8 = 8×F ÷ 8 which makes ⅝×C = F
Hint #8
Substitute (⅝×C) for F in eq.5b: 5×B = 3×(⅝×C) which is equivalent to 5×B = 1⅞×C Divide both sides of the equation above by 5: 5×B ÷ 5 = 1⅞×C ÷ 5 which makes B = ⅜×C
Hint #9
Substitute (⅝×C) for F, and ⅜×C for B in eq.6a: 2×(⅝×C) – ⅜×C = D which becomes 1¼×C – ⅜×C = D which means ⅞×C = D
Hint #10
Substitute (⅜×C) for B in eq.2a: 2×(⅜×C) = A which makes ¾×C = A
Solution
Substitute ¾×C for A, ⅜×C for B, ⅞×C for D, 0 for E, and ⅝×C for F in eq.1: ¾×C + ⅜×C + C + ⅞×C + 0 + ⅝×C = 29 which simplifies to 3⅝×C = 29 Divide both sides of the equation above by 3⅝: 3⅝×C ÷ 3⅝ = 29 ÷ 3⅝ which means C = 8 making A = ¾×C = ¾ × 8 = 6 B = ⅜×C = ⅜ × 8 = 3 D = ⅞×C = ⅞ × 8 = 7 F = ⅝×C = ⅝ × 8 = 5 and ABCDEF = 638705