# Four Birds + One

You have a 7x7 tray, and several pieces as shown (the dimensions should be fairly obvious since the picture is to scale, but if not, a yellow bird piece fits snugly in a 4x4 square, the blue piece is a 2x2 square, and the red piece fits snugly in a 3x3 square).

Challenge 1: Fit the four yellow birds into the tray, no overlapping. Rotation and reflection are allowed.

Challenge 2: Fit the four yellow birds and the blue piece into the tray, no overlapping. Rotation and reflection are allowed.

Challenge 3: Fit the four yellow birds and the red piece into the tray, no overlapping. Rotation and reflection are allowed.

Challenge 4: One of the above challenges is impossible, but you probably solved it anyway. Determine which one is impossible, and also determine the smallest size square that you can pack that set of pieces into.

• What does it mean "but you probably solved it anyway?" – Oliver Ni Dec 24 '16 at 4:01
• That's up to you to discover :) In any case, I wouldn't worry about challenge 4 until you've solved at least one of 2 or 3. – TheGreatEscaper Dec 24 '16 at 4:02
• Are we allowed to rotate and reflect the tray? (This is a different question in disguise) – humn Dec 24 '16 at 4:15
• @humn so long as the pieces all fit into the tray without overlapping, then it's all fair game! – TheGreatEscaper Dec 24 '16 at 4:18
• @Nice :) I've been following your updates, and I think you 'solved' the challenge before realising the trick... 'you probably solved it anyway' turned out to be quite accurate, no? Anyway, have fun with the last part left to solve! – TheGreatEscaper Dec 24 '16 at 4:32

Challenge 1: Fit the four yellow birds into the tray, no overlapping. Rotation and reflection are allowed.

Challenge 2: Fit the four yellow birds and the blue piece into the tray, no overlapping. Rotation and reflection are allowed.

If we configure it as follows and rotate it exactly 72 degrees counterclockwise, it comes out to exactly 699/700px. (Thanks @Deusovi!)

Oh my god this makes the whole canvas look tilted.

Challenge 3: Fit the four yellow birds and the red piece into the tray, no overlapping. Rotation and reflection are allowed.

Impossible. See Challenge 4.

Challenge 4: One of the above challenges is impossible, but you probably solved it anyway. Determine which one is impossible, and also determine the smallest size square that you can pack that set of pieces into.

The impossible one is Challenge 3. It looks like it can be solved:

But hmm... wait a second. You can also see in the image that they are overlapping! Each diagonal is $\sqrt{2}$, so the smallest square it can fit in is one of side length $5 * \sqrt{2} = 7.07106781$!

Sorry guys, I was too lazy to save and export all the images, so I just took screenshots. xD