Puzzle for March 24, 2021 ( )
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 A to both sides of eq.5: F – A – E + A = A + C + A which becomes eq.5a) F – E = 2×A + C Add A, C, and F to both sides of eq.2: B – A – C + A + C + F = A – F + A + C + F which simplifies to eq.2a) B + F = 2×A + C
Hint #2
In eq.5a, replace 2×A + C with B + F (from eq.2a): F – E = B + F In the equation above, subtract F from both sides, and add E to both sides: F – E – F + E = B + F – F + E which becomes 0 = B + E Since B and E are non-negative, the equation above makes: B = 0 and E = 0
Hint #3
In eq.3, replace E and B with 0: C + D – 0 = A + 0 + F which becomes eq.3a) C + D = A + F
Hint #4
In eq.4, replace E with 0: D + 0 = C + F which becomes eq.4a) D = C + F
Hint #5
In eq.3a, substitute C + F for D (from eq.4a): C + C + F = A + F which becomes 2×C + F = A + F Subtract F from each side of the above equation: 2×C + F – F = A + F – F which makes 2×C = A
Hint #6
Substitute 0 for B, and (2×C) for A in eq.2a: 0 + F = 2×(2×C) + C which becomes F = 4×C + C which makes F = 5×C
Hint #7
Substitute 5×C for F in eq.4a: D = C + 5×C which makes D = 6×C
Solution
Substitute 2×C for A, 0 for B and E, 6×C for D, and 5×C for F in eq.1: 2×C + 0 + C + 6×C + 0 + 5×C = 14 which simplifies to 14×C = 14 Divide both sides of the equation above by 14: 14×C ÷ 14 = 14 ÷ 14 which means C = 1 making A = 2×C = 2×1 = 2 D = 6×C = 6×1 = 6 F = 5×C = 5×1 = 5 and ABCDEF = 201605