Puzzle for August 11, 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
Add D and B to both sides of eq.4: C – D + D + B = A – B + D + B which becomes C + B = A + D which may be written as B + C = A + D In the above equation, replace B + C with D + E (from eq.2): D + E = A + D Subtract D from both sides: D + E – D = A + D – D which makes E = A
Hint #2
In eq.5, replace B + C with D + E (from eq.2): E + F = A + D + E Subtract E from both sides of the equation above: E + F – E = A + D + E – E which becomes eq.5a) F = A + D
Hint #3
In eq.3, substitute A + D for F (from eq.5a): B + D = A + A + D which becomes B + D = 2×A + D Subtract D from each side of the above equation: B + D – D = 2×A + D – D which makes B = 2×A
Hint #4
Substitute A for E, A + D for F (from eq.5a), and 2×A for B in eq.6: A + A – D + A + D = C – 2×A which becomes 3×A = C – 2×A Add 2×A to both sides of the equation above: 3×A + 2×A = C – 2×A + 2×A which makes 5×A = C
Hint #5
Substitute A for E, 2×A for B, and 5×A for C in eq.2: D + A = 2×A + 5×A which becomes D + A = 7×A Subtract A from both sides of the equation above: D + A – A = 7×A – A which makes D = 6×A
Hint #6
Substitute 6×A for D in eq.5a: F = A + 6×A which makes F = 7×A
Solution
Substitute 2×A for B, 5×A for C, 6×A for D, A for E, and 7×A for F in eq.1: A + 2×A + 5×A + 6×A + A + 7×A = 22 which simplifies to 22×A = 22 Divide both sides of the above equation by 22: 22×A ÷ 22 = 22 ÷ 22 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 5×A = 5 × 1 = 5 D = 6×A = 6 × 1 = 6 E = A = 1 F = 7×A = 7 × 1 = 7 and ABCDEF = 125617