Puzzle for June 30, 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.3, replace C with B + D (from eq.2): D + F = A + B + D + E Subtract D from both sides of the above equation: D + F – D = A + B + D + E – D which becomes eq.3a) F = A + B + E
Hint #2
In eq.3a, replace B + E with F – E (from eq.4): F = A + F – E In the equation above, add E to both sides, and subtract F from both sides: F + E – F = A + F – E + E – F which makes E = A
Hint #3
In eq.3, substitute A + F for C (from eq.5), and A for E: D + F = A + A + F + A Subtract F from each side of the equation above: D + F – F = A + A + F + A – F which makes D = 3×A
Hint #4
Substitute A for E, and 3×A for D in eq.6: B – A = A + 3×A + A which becomes B – A = 5×A Add A to each side of the above equation: B – A + A = 5×A + A which makes B = 6×A
Hint #5
Substitute 6×A for B, and A for E in eq.3a: F = A + 6×A + A which makes F = 8×A
Hint #6
Substitute 8×A for F in eq.5: C = A + 8×A which makes C = 9×A
Solution
Substitute 6×A for B, 9×A for C, 3×A for D, A for E, and 8×A for F in eq.1: A + 6×A + 9×A + 3×A + A + 8×A = 28 which simplifies to 28×A = 28 Divide each side of the equation above by 28: 28×A ÷ 28 = 28 ÷ 28 which means A = 1 making B = 6×A = 6 × 1 = 6 C = 9×A = 9 × 1 = 9 D = 3×A = 3 × 1 = 3 E = A = 1 F = 8×A = 8 × 1 = 8 and ABCDEF = 169318