Picture of Colin Ni

Colin Ni

Emailcolinni@math.ucla.edu
About Me

I'm a fifth-year graduate student at the UCLA Department of Mathematics. My advisor is Paul Balmer. I ATC'ed Spring 2024 (here are slides), and I passed the algebra and geometry/topology quals in Fall 2021 and the basic in Spring 2022 (kekw).

I play blitz chess, and I used to play live high-stakes poker. I interned at HRT this past summer, and before grad school I worked at Scale AI as a machine learning research engineer. Before that, I did my undergrad at the College of Creative Studies (CCS) at UCSB, where I took these classes.
Teaching History
50 Fun Puzzles (sorted by difficulty)
  1. Color the plane using two colors. Are there always two points 1 distance apart with the same color?
  2. Two players take turns placing coins onto a round table without overlap, and the last person who places a coin wins. Who wins?
  3. How many cuts does it take to cut a 3x3x3 block of tofu into 27 1x1x1 pieces? You are allowed to move pieces around between cuts.
  4. Some ants are walking on a log, all at the same speed. When two ants bump into each other, they turn around and walk the other way, and when they reach the end of the log, they fall off. How long will it take before they all fall off (if ever)?
  5. A hundred coins are on a table in front of you, exactly 10 of which show heads, but you are blindfolded. Can you split the coins, flipping them as you desire, into two groups that contain the same number of heads?
  6. An 8x8 chess board is coverable by 32 2x1 dominoes. If you remove the bottom-left and top-right corners, is it coverable by 31 2x1 dominoes?
  7. A crocodile has n teeth, one of which is sore and will make the crocodile bite if pulled. You and your n - 1 friends decide to each pull a tooth, one by one. Would you rather be first, second, ..., last?
  8. Show that the set of rational numbers Q has measure 0, i.e. show that you can cover Q by a collection of line segments whose total length is arbitrarily small.
  9. On an island lives some cannibals. Two cannibals who are awake will not try to eat each other, but if a cannibal eats a person, then they fall asleep and are thus vulnerable to getting eaten. A visitor who is not a cannibal visits the island. Is the visitor safe?
  10. Show that there exists arbitrarily large gaps between primes.
  11. Four coins are on a spinning platform in front of you, but you are blindfolded. On each turn you may flip any number of the coins you want, but after every turn someone gets to rotate the platform any way they want. Can you make them all heads or all tails in a finite number of turns?
  12. Your friend thinks of a polynomial p with natural number coefficients, and your job is to determine p by asking for the value of p at natural numbers. Can you do with finitely many questions, and if so, how many?
  13. Show that on a finite-dimensional vector space, a linear operator is invariant on all hyperplanes only if it is a scalar multiple of the identity.
  14. In a tribe lives some villagers. They all have red eyes but are forbidden to speak of it because the island is cursed: if you fall asleep knowing you have red eyes, you die. One day a foreigner lets it slip that someone has red eyes. What becomes of the villagers?
  15. Suppose the (complex) character table of a finite group contains the row (1, w, w^2, 1), where w is a primitive 3rd root of unity. What can the group be?
  16. A mouse is stuck in a circular pond, and a cat is at the edge of the pond, trying to catch the mouse. The cat can run four times as fast as the mouse can swim, but the mouse can outrun the cat. The cat cannot swim. Can the mouse escape the pond?
  17. Show that a finite group acting transitively on a non-singleton set has a fixed-point-free element.
  18. Given an n by n matrix of nonnegative numbers and a threshold t, find an efficient algorithm to determine the maximum k such that any k by k contiguous submatrix has sum at most t.
  19. Show that you can cover any 10 points in the plane with disjoint unit disks.
  20. Let p > 3 and p + 2 be twin primes. Add up the digits of p(p+2), add up the digits of that resulting number, and so on, until you get a one-digit number. Show that this one-digit number is 8.
  21. Let B be the nxn matrix with ij-entry given by the number of shared divisors of i and j. What is the determinant of B?
  22. Settlers discover an infinite grid and decide to create a city. On day 1, they build a finite number of height 1 buildings. On day n they build a new building of height n such that the heights of the surrounding (eight) buildings sum to n, but they stop building altogether if this is impossible. Does the city ever stop expanding?
  23. Two players each choose a positive integer. You pay the other player $1 if your number is exactly one less than theirs, but you get paid $2 if your number is more than one less than theirs. What is each player's strategy?
  24. A safe has 30 lights on it. Once per minute, you may flip any of the lights on or off and then try opening the safe; if all 30 lights are correct, then the safe opens, but otherwise the safe indicates whether you got at least half of the lights correct. How quickly can you open this safe, on average?
  25. Show that a square matrix with positive entries has a positive eigenvalue.
  26. There is a line of lilypads, one for each integer, and a flea is hopping on the lilypads, moving once per second according to an arithmetic progression. A frog, which can jump to any lilypad once per second is trying to catch the flea. Can the frog always succeed?
  27. Is 420 a square mod the Mersenne prime 2^127 - 1? What about 421?
  28. A lion is trying to catch one of 100 sheep in the plane R^2. Once the sheep decide on a starting position, each turn the lion moves in any direction up to 10 meters and then so does one of the sheep. Can the lion catch a sheep?
  29. Construct a bijection between (0, 1) and [0, 1).
  30. What is the probability that a random inscribed n-gon contains the center of the circle?
  31. Your friend chooses a nontrivial probability distribution, samples two distinct real numbers from it, and randomly tells you one of them. Can you guess whether the number you were told was larger than the other one and be correct more than half the time?
  32. Does there exist a polynomial in two variables with real coefficients whose range is the positive real numbers?
  33. Can you solve a 4x4 sliding tile puzzle where the 14 and 15 are swapped?:
      1   2   3   4
      5   6   7   8
      9   10  11  12
      13 15  14  []
  34. A hundred prisoners are numbered 1-100, and release forms numbered 1-100 for the prisoners are placed randomly in 100 drawers, one in each drawer. If all prisoners find their release forms, they will all be released. However, each prisoner may check only 50 drawers, and they are forbidden to communicate after the process begins. Can they do better than (1/2)^100?
  35. For which positive integers n are cos(2pi/n) rational?
  36. Show that if a smooth compact connected oriented manifold without boundary has Euler characteristic zero, then it admits a nowhere vanishing vector field.
  37. Fix a finite number of vectors in a finite-dimensional vector space over an infinite field. Show that there exists a functional that does not vanish on any of these vectors.
  38. Three blind spiders and a crippled fly walk along the edges of a regular tetrahedron. The spiders can all walk faster than the fly but cannot see it. Can the spiders catch the fly?
  39. Show that in any subset of 1, 2, ..., 200 with size 101, there exists two distinct numbers where one divides the other.
  40. You and an accomplice steal a necklace that has jewels of n different colors, with an even number of jewels of each color. Show that by cutting the necklace at n + 1 spots, you can split the jewels of each color evenly with your accomplice.
  41. Given a p-faced die for each p = 2, 3, 5, ..., 41, how can you generate a uniformly random integer in [1, 42] in the fewest number of rolls (on average, if non-deterministic)?
  42. Estimate the number of hydrogen atoms in the sun (within 2 orders of magnitude).
  43. Can you partition R^2 into circles? What about R^3?
  44. A herd of 101 cows has the property that if any cow is removed from the herd, then the remaining 100 cows can be split into two groups of equal weight and equal size. Show that the cows all have the same weight.
  45. Given a number, you may transform the number by inserting plusses and then evaluating the sum; for example, you may transform 822 into 8+2+2=12 or 82+2=84 or 8+22=30. Can you trasform any number into a one-digit number in 4 turns?
  46. Show that for every n there is a real extension of the rationals with Galois group Z/n.
  47. Prisoners 1, 2, 3, ... are each wearing a hat with a real number on it and are lined up so that prisoner n can only see prisoner n + k for every positive k. The prisoners are asked to guess the number on their hat, being set free if they guess their number correctly, and they are given a chance to strategize beforehand. How many prisoners can be set free if they must all guess at the same time? What if the numbers are all 0 or 1 and they guess one by one, starting with prisoner 1?
  48. Show that any smooth curve in the plane contains four points that form a rectangle.
  49. A robot travels on a planar graph in the following way. It starts on the interior of a random edge and travels along it until it reaches a vertex. Then it rotates counter-clockwise until it is aligned with an edge leaving the vertex, and it travels along that edge, and so on. Show that given a finite number of points in the plane, no three of which are colinear, there exists a planar graph on these vertices such that the robot will visit every vertex infinitely many times no matter its starting position.
  50. An urn initially contains one red ball and one blue ball, and we repeatedly duplicate a random ball in the urn. What is the limiting distribution of the proportion of red balls?
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