I gave Abstract pacman a go today.

I am rusty in trigonometry and I came up with all sorts of solutions to this problem but was too lazy too look up whether I remembered trig laws correctly and came up with the simplest thing i could.

I started with the assumption that the the intersection point of the two circles would be such that the angle to that Point of Intersection would be \(\frac{\pi}{4}\) radians from the origin's of both circles. This was a bit of intuition as to what yields maximum area, the image helped. I also assumed the circle intersected with both axis.

That means if it's in the 4th (lower right/negative y,positive x) quadrant the POI would be (\(2\sqrt{2},-2\sqrt{2})\)

Let's define the inner circle's radius as \(r_{2}\).

That means looking at the inner circle you could build a triangle from the origin to (\(2\sqrt{2}\),\(2\sqrt{2}\)). since the angle is 45° it is and isosceles with \(r_{2}\) as the hypotenuse and \(\frac{r_{2}}{\sqrt{2}}\) as the two equal sides.

Since we assumed the circle intersected with the y axis we can build the relation that \(r_{2} \) + \(\frac{r_{2}}{\sqrt{2}} = 2\sqrt{2}\)

we can now solve for \(r_{2}\) as we have \[r_{2} (1+ \frac{1}{\sqrt{2}}) = 2\sqrt{2}\]

I screwed up by calculating \(\frac{2\sqrt{2}}{1+\frac{2}{\sqrt{2}}}\) when i should have done \(\frac{2\sqrt{2}}{1+\frac{1}{\sqrt{2}}}\) and got ~1.17157 or 1.172 rounded up to 3 digits instead of 1.657.

I also forgot to double for diameter which results in 3.314 as the final answer.

My point is let anyone post a solution maybe just flag or differentiate ones that come from people who didn't solve correctly. I haven’t checked whether anyone posted this exact solution yet but realistically this could have happened to someone else with a unique solution who failed to submit a correct answer or took a peek at solutions.

## Comments

Sort by:

TopNewest@Bug Menot We do not allow people who got a problem wrong to post a solution, as it leads to a lot of wrong solutions posted and further confusion. If you desire, you can post your solution as a report, and request for it to be converted into a solution. – Calvin Lin Staff · 7 months, 3 weeks ago

Log in to reply

One of the problems with letting anyone post solutions is that we'd end up with too much spam, making mostly irrelevant comments.

However, if you do realize that you have some important insight to the problem, you could write up your ideas and submit them as a comment, the staff will convert them to solutions – Agnishom Chattopadhyay · 7 months, 3 weeks ago

Log in to reply

Also I said differentiate it. maybe have a hidden/obscured section for them? It's not a big deal and my proposal isn't that well thought out but from a user perspective it was a bit disappointing I couldn't submit anything. – Bug Menot · 7 months, 3 weeks ago

Log in to reply