Puzzle for August 15, 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 positive integer.
Scratchpad
Help Area
Hint #1
Add C and D to both sides of eq.3: B – C + C + D = A – D + C + D which becomes B + D = A + C In eq.5, replace B + D with A + C: A + C = C + E Subtract C from both sides of the above equation: A + C – C = C + E – C which makes A = E
Hint #2
Add D and E to both sides of eq.4: F – D + D + E = B – E + D + E which becomes F + E = B + D In eq.5, replace B + D with F + E: F + E = C + E Subtract E from both sides of the above equation: F + E – E = C + E – E which makes F = C
Hint #3
In eq.2, substitute C for F, and A for E: C + C = A + A which makes 2×C = 2×A Divide both sides of the above equation by 2: 2×C ÷ 2 = 2×A ÷ 2 which means C = A and also means F = C = A = E
Hint #4
Substitute F for A, C, and E in eq.6: F × B = (F × F) + F which may be written as F × B = F × (F + 1) Divide both sides of the equation above by F: F × B ÷ F = F × (F + 1) ÷ F which means eq.6a) B = F + 1
Hint #5
Substitute F + 1 for B (from eq.6a), and F for C and E in eq.5: F + 1 + D = F + F Subtract F and 1 from both sides of the above equation: F + 1 + D – F – 1 = F + F – F – 1 which means eq.5a) D = F – 1
Solution
Substitute F for A and C and E, F + 1 for B (from eq.6a), and F – 1 for D (from eq.5a) in eq.1: F + F + 1 + F + F – 1 + F + F = 42 which simplifies to 6×F = 42 Divide both sides of the equation above by 6: 6×F ÷ 6 = 42 ÷ 6 which means F = 7 making A = C = E = F = 7 B = F + 1 = 7 + 1 = 8 (from eq.6a) D = F – 1 = 7 – 1 = 6 (from eq.5a) and ABCDEF = 787677