Solstice Puzzles ANSWERS 

Laterally Yours

1: Turn the 9 upside down (columns add up to 21). 2: Place the 1 on top of the 5 (20). 3: Place the 3 on the 9 (18). 4: Place the 5 on the 7 (19). 5: Place the 1 facedown on the 4 (20). 6: Place the 4 facedown on the 1 (20). 7: Place the 3 facedown on the 6 (18). 8: Place the 6 facedown on the 3 (18). 9: Place the 2 facedown on the 5 (19). 10: Place the 5 facedown on the 2 (19). 11: Place the 3 partially under the 2, to make 2 cubed (24). 12: Take any card, set fire to it and use it to set fire to the other cards (0).

Any more ideas?

Six Weeks of Seconds

Same, 3,628,800, since both factor out to 2⁸ x 3⁴ x 5² x 7¹. (For what it's worth, the number of milliseconds in a day is 5⁵ x 4⁴ x 3³ x 2² x 1¹).

Squares

10. Let a and b = lengths of the sides of the large and small yellow squares respectively. Then area of large red square = (a+b)², and of small red square (a-b)², so (a+b)² + (a-b)² = 2(a² + b²) = 20. So a² + b² = sum of areas of yellow squares = 10.

Instant Arithmetic

9,999, since a² – b² = (a + b) (a – b) = (9,999) (1)

The Prize

Green triangle. Round 1: If it had been under the blue square or yellow circle, one of them would have said so. Round 2: Having eliminated the blue square and yellow circle, Alice learns it's not under the blue triangle (or Bob, knowing the color, would have said so), while Bob learns it's not under the green circle (or Alice, knowing the shape, would have said so). Round 3: Alice knows that Bob would have said if the prize was under the blue triangle, and Bob knows that Alice would have said if the prize was under the green circle. So now they both know it's under the green triangle.

Joining

I read somewhere that this is easier to solve on the second day, so try again tomorrow.