The Lucky Tickets Problem Noor Doukmak February 19, 2013 In Russia… 834012 Bus Bcket Lucky Tickets… 942807 100100 230113 740623 123411 What makes them lucky? 004013 Lucky Tickets Problem Alexandre Kirillov in the early 70’s •  How many lucky Bckets exist? •  We would like to count the number of Bckets d1d2d3d4d5d6 for which d1+d2+d3=d4+d5+d6 More generally, •  For 2r-­‐digit numbers in base q, Lucky Sum •  Each lucky Bcket has a lucky sum n: Let an be the number of r-­‐digit numbers whose digits sum to n: n 0 1 2 … 27 d1d2d3 an 000 1 3 6 100, 010, 001 200, 020, 002, 101, 110, 011 … 999 … 1 Number of lucky Bckets with lucky sum n is number of lucky Bckets in general is The Let Claim L(q,r) is the sum of the squares of the coefficients of Aqr(s) Proof Conclusion •  In order to find L(q,r), we must find the coefficients of the polynomial To solve our problem… •  We want to find •  Thus, we need to know a0 , a1 , a2 , a3 , … a27 which are the coefficients of Two steps 55252 55252 = 5.5252% 106 References •  Lando, S. K. "1.1 The Lucky Tickets Problem." Lectures on Genera8ng Func8ons. Providence, RI: American MathemaBcal Society, 2003. 1-­‐3. Print. •  Jonathon Novak, MIT (Notes) Thank you. QuesBons?