October 13

•  Problems with remainders
• $39^{192}\mod 11$;
• $101^{1001}\mod 7$;
• $a^6\mod 7$ for $a=1,2,\ldots,6\$.
• Tetrahedra and pyramids.
• The skeleton of a tetrahedron made out of wire. Play the game: cut an edge, in turns, until it falls apart. Who wins?
• Same for pyramid with square base.
• To a triangular face of a regular (solid) square pyramid one glues on the regular solid tetrahedron. How many faces the resulting solid body has?
• More experiments with Polish notation:
• $+++++++1,2,3,4,5,6,7,8$.
• Does the answer depend on the order of numbers?
• What about $+1+2+3+4+5+6+7,8$?
• One can get different trees - can one draw all of them?
• The following sets are given: $$A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\}, D = \{0, 7, 100, 333\}.$$
  Describe the sets:
• $A \cup B$;
• $A\cap B$;
• $(A \cap B)\cup D$;
• $C \cap (D\cap B)$;
• $(A\cup B)\cap (C \cup D)$;
• $(A\cap B)\cup (C \cap D)$;
• $(D\cup A)\cap (C \cup B)$;
• $(A\cap (B\cap C))\cap D$;
• $(A\cup (B\cap C))\cap D$;
• $(C \cap A)\cup ((A\cup (C \cap D))\cap B)$
• Origami!
• Game: bulls and cows