Another is "_, _, _, _, _, _, 35, _". Then $(3p + aq, 5p + bq) = (0, 1)$, which means $$3 = 3(1) - 5(0) = 3(5p+bq) - 5(3p+aq) = (5a-3b)(-q). The logic is this: the blanks before 8 include 1, 2, 4, and two other numbers. Barbra made a clay sculpture that has a mass of 92 wants to make a similar... (answered by stanbon).
The sides of the square come from its intersections with a face of the tetrahedron (such as $ABC$). If it's 5 or 7, we don't get a solution: 10 and 14 are both bigger than 8, so they need the blanks to be in a different order. How do we use that coloring to tell Max which rubber band to put on top? We either need an even number of steps or an odd number of steps. With an orange, you might be able to go up to four or five. Misha has a cube and a right square pyramid area formula. For example, the very hard puzzle for 10 is _, _, 5, _. There is also a more interesting formula, which I don't have the time to talk about, so I leave it as homework It can be found on and gives us the number of crows too slow to win in a race with $2n+1$ crows. Actually, we can also prove that $ad-bc$ is a divisor of both $c$ and $d$, by switching the roles of the two sails.
There's a lot of ways to explore the situation, making lots of pretty pictures in the process. We can cut the 5-cell along a 3-dimensional surface (a hyperplane) that's equidistant from and parallel to edge $AB$ and plane $CDE$. If, in one region, we're hopping up from green to orange, then in a neighboring region, we'd be hopping down from orange to green. Finally, one consequence of all this is that with $3^k+2$ crows, every single crow except the fastest and the slowest can win. Misha has a cube and a right square pyramid surface area calculator. If $2^k < n \le 2^{k+1}$ and $n$ is odd, then we grow to $n+1$ (still in the same range! ) That we cannot go to points where the coordinate sum is odd.
For 19, you go to 20, which becomes 5, 5, 5, 5. Notice that in the latter case, the game will always be very short, ending either on João's or Kinga's first roll. If we draw this picture for the $k$-round race, how many red crows must there be at the start? How many such ways are there? If you cross an even number of rubber bands, color $R$ black.