Puzzle for August 27, 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.3, replace A + E with D (from eq.4): B + C + D = D + F Subtract D from both sides of he above equation: B + C + D – D = D + F – D which becomes eq.3a) B + C = F
Hint #2
In eq.2, substitute (B + C) for F (from eq.3a): A + C – (B + C) = E – B which is equivalent to A + C – B – C = E – B which becomes A – B = E – B Add B to both sides of the above equation: A – B + B = E – B + B which makes A = E
Hint #3
In eq.4, substitute A for E: D = A + A which means D = 2×A
Hint #4
Substitute A for E, and 2×A for D in eq.6: A – A = 2×A – B which becomes 0 = 2×A – B Add B to both sides of the above equation: 0 + B = 2×A – B + B which means B = 2×A and which also means B = D = 2×A
Hint #5
Substitute 2×A for B, and A for E in eq.5: F = A + 2×A + A which makes F = 4×A
Hint #6
Substitute 2×A for B and D, A for E, and 4×A for F in eq.3: 2×A + C + 2×A = A + A + 4×A which becomes 4×A + C = 6×A Subtract 4×A from both sides of the above equation: 4×A + C – 4×A = 6×A – 4×A which makes C = 2×A
Solution
Substitute 2×A for B and C and D, A for E, and 4×A for F in eq.1: A + 2×A + 2×A + 2×A + A + 4×A = 24 which simplifies to 12×A = 24 Divide both sides of the above equation by 12: 12×A ÷ 12 = 24 ÷ 12 which means A = 2 making B = C = D = 2×A = 2 × 2 = 4 E = A = 2 F = 4×A = 4 × 2 = 8 and ABCDEF = 244428