I think you're gonna lose your bet. Here's why:
Let's start by finding the solution to the quadritaisho (although there are only three symbols in this puzzle. Does that make it just a tritaisho? A traisho? We can come back to that later). Here are all of the basic deductions we can make by drawing lines between symbols and using the symbols' symmetry rules:
Let's look over on the left. The fourth cell down, immediately below the 1x1 room with the black circle, now has to be connected to the white circle below it. Using the 180° symmetry rule, we now know that that room has to be four cells tall. This now creates two borders on the white circle below that one, and one border on the white diamond to the right, allowing us to complete those rooms:
Next, look at the bottom two rows. We already have the middle 2x2 assigned to the white diamond, and we have two horizontal bars straddling the line on either side. On the left side, it's very easy to see that all of the remaining cells must be assigned to the white bar, as not doing so would break the symmetry of the black bar above it. On the right, the black circle can technically still invade the penultimate column without breaking symmetry rules, but the cells immediately below it must be part of the room containing the white bar, and in order to get there while maintaining symmetry, it must use both rows:
Continuing in the bottom right, consider the cell immediately above the white bar. It has four symbols it can reach, but only one of them will work: the black circle to its immediate left is blocked on the other side by the black diamond, the white bar would have to invade the white circle in order to be reflectionally symmetric, and the other black circle runs into a white circle at the rotationally symmetric point. Therefore, that cell has to be part of the room with the white circle on the far right, allowing us to complete that room. We also now have three sides to the room with the lowermost black circle, so we can finish that off as well:
Jumping to the top right, because I think it's the most fun. The cell all the way in the top right is going to be hard to get to. It can actually be reached by two symbols. First, we have the nearby white circle, which can reach it using this Loch Ness monster of a room without actually breaking anything:
But it's pretty clear that the cell marked in red there can no longer be visited by anything, so that's out. The only other symbol that can reach the top right is the horizontal bar all the way on the left, which of course means that that room needs to consist of the entirety of the first two rows:
Now the middle of the grid starts to fall apart. Thanks to that big ol' line we just drew, the black circle in r3c3 can't escape downward, meaning the cell below it must belong to the white bar in row 5. That almost closes off the rooms below it and to its left, which can be closed off, and that starts a chain reaction of black-circle rooms, starting with the aforementioned one and proceeding down and to the right until we get here:
Almost done. Now, note that the bottom-right black circle has complete top and right boundaries, so we can close that off. This means that the cell to its immediate left can now only be reached by the other black circle, creating a Z-shaped room. All that's left to do is to close off the path between the white circle and horizontal bar, completing our puzzle!
Now, as requested, let's shade in the rooms with black symbols:
This does look awfully familiar...
Any self-respecting, card-carrying American sports fan should recognize this as the logo of the New York Giants:
At this point, it's perfectly reasonable to be confident in your guess. However,
it's also perfectly reasonable for your friend to be confident that you're wrong, because naturally, the New York Giants do not play in New York City, nor even New York State, but rather, in East Rutherford, New Jersey.
Hope you didn't wager too much. Always bet responsibly, folks!
