Puzzle for August 7, 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
In eq.4, replace B + C with F (from eq.3): eq.4a) A + F = E
Hint #2
In eq.5, replace E with A + F (from eq.4a): A + F – B = F – D Subtract F from both sides of the above equation: A + F – B – F = F – D – F which becomes A – B = –D Add B and D to both sides: A – B + B + D = –D + B + D which becomes eq.5a) A + D = B
Hint #3
In eq.2, replace A + D with B (from eq.5a): C = B
Hint #4
In eq.3, substitute C for B: F = C + C which makes F = 2×C
Hint #5
Substitute C for B, and 2×C for F in eq.6: C × D = C + 2×C which makes C × D = 3×C Divide both sides of the above equation by C: C × D ÷ C = 3×C ÷ C which makes D = 3
Hint #6
Substitute 3 for D in eq.2: C = A + 3 Subtract 3 from each side: C – 3 = A + 3 – 3 which makes eq.2a) C – 3 = A
Hint #7
Substitute C – 3 for A (from eq.2a), and C for B in eq.4: C – 3 + C + C = E which becomes eq.4a) 3×C – 3 = E
Solution
Substitute C – 3 for A (from eq.2a), C for B, 3 for D, 3×C – 3 for E (from eq.4a), and 2×C for F in eq.1: C – 3 + C + C + 3 + 3×C – 3 + 2×C = 29 which simplifies to 8×C – 3 = 29 Add 3 to both sides of the above equation: 8×C – 3 + 3 = 29 + 3 which makes 8×C = 32 Divide both sides by 8: 8×C ÷ 8 = 32 ÷ 8 which means C = 4 making A = C – 3 = 4 – 3 = 1 (from eq.2a) B = C = 4 E = 3×C – 3 = 3×4 – 3 = 12 – 3 = 9 (from eq.4a) F = 2×C = 2×4 = 8 and ABCDEF = 144398