Puzzle for November 4, 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
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Hint #1
In eq.1, replace A + D with E + F (from eq.4): B = E + F + E which becomes eq.1a) B = 2×E + F
Hint #2
In eq.2, replace B with 2×E + F (from eq.1a): F = 2×E + F – D In the equation above, subtract F from both sides, and add D to both sides: F – F + D = 2×E + F – D – F + D which makes D = 2×E
Hint #3
Add D to both sides of eq.2: F + D = B – D + D which becomes F + D = B Substitute C + E for F (from eq.3) in the above equation: eq.2a) C + E + D = B
Hint #4
Substitute C + E + D for B (from eq.2a) in eq.1: C + E + D = A + D + E Subtract D and E from both sides of the above equation: C + E + D – D – E = A + D + E – D – E which simplifies to C = A
Hint #5
Substitute A for C in eq.5: E = A ÷ A which makes E = 1 making D = 2×E = 2 × 1 = 2
Hint #6
Substitute 2 for D, and 1 for E in eq.1: B = A + 2 + 1 which makes eq.1b) B = A + 3
Hint #7
Substitute 2 for D, A + 3 for B (from eq.1b), and 1 for E in eq.6: A × 2 = A + 3 + 1 which becomes 2×A = A + 4 Subtract A from both sides of the equation above: 2×A – A = A + 4 – A which makes A = 4 and also makes C = A = 4
Hint #8
Substitute 4 for A in eq.1b: B = 4 + 3 which makes B = 7
Solution
Substitute 7 for B, and 2 for D in eq.2: F = 7 – 2 which makes F = 5 and makes ABCDEF = 474215