The original question includes a diagram: A square spiral drawn on axes running from −5 to 5, winding anticlockwise outwards, with an arrow on the outermost segment showing the direction of travel.
A spiral line is drawn as shown. Starting from the point (−2, 0) it travels down to (−2, −2), right to (2, −2), up to (2, 2), left to (−4, 2), down to (−4, −4), right to (4, −4), up to (4, 4) and then left along the line y = 4.
This spiral pattern continues indefinitely.
Which one of the following points is not on the spiral line?
- A(99, 100)
- B(99, −100)
- C(−99, 100)
- D(−99, −100)
- E(100, 99)
- F(100, −99)
- G(−100, 99)
- H(−100, −99)
Show the answer and worked solution
answer · G
- A(99, 100)
- B(99, −100)
- C(−99, 100)
- D(−99, −100)
- E(100, 99)
- F(100, −99)
- G(−100, 99)
- H(−100, −99)
Describe the four families of segments. For each n ≥ 1 the spiral contains the right edge x = 2n with −2n ≤ y ≤ 2n, then the top edge y = 2n with −(2n+2) ≤ x ≤ 2n, then the left edge x = −(2n+2) with −(2n+2) ≤ y ≤ 2n, then the bottom edge y = −(2n+2) with −(2n+2) ≤ x ≤ 2n+2. Notice the asymmetry: each left edge stops at height 2n, two units short of the top edge that sits at 2n + 2. Every option has one even coordinate, so check it against the matching segment. The point (−100, 99) needs the left edge x = −100, which comes from n = 49 and runs only up to y = 98, so 99 is off the end. Every other option lies inside its segment.