Looking for U.K. problem column (?) from 1980s

While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher way back when. Four of the problems are labeled “Set U” and the other four are labeled “Set V”. I’m reproducing “Set U” below, … Read more

How many arrangements of nn points with kk edge lengths exist in dd dimensions?

[Asking on behalf of a high school mathematics course, but responses written at any level are welcome!] I was recently reading over a nice puzzle called the four points, two distances problem: Find all the ways to arrange 4 points so that only 2 distances occur between any two points. The author of this piece, … Read more

Finding roots of equation with gamma functions

Encountered this function in one of my research problems Γ(1−ica−γ)Γ(1+ica+N2−γ)Γ(1+ica−γ)Γ(1−ica+N2−γ)−(aγ+b−ic2)(aγ+b−ic2−1aγ+b+ic2−1)Naγ+b+ic2=0where {a,b,c}∈C and N∈Z+ I am wondering if it is possible to solve for γ analytically. I solved it numerically for various parameters and the answers are interesting (γ′s lie on some sort of nice curve). For example for N→20,a→1.53−0.02i,b→0.098−0.3i,c→−0.3+2i solutions look like and for another set of random … Read more

Help me find good math questions for my students [closed]

Closed. This question is opinion-based. It is not currently accepting answers. Want to improve this question? Update the question so it can be answered with facts and citations by editing this post. Closed 4 years ago. Improve this question I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad. Now … Read more

Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling: “This exponential behavior makes solving polynomial systems difficult and explains why there are few solvers that are able to automatically solve systems with Bézout’s bound higher … Read more

Covering the disk with a family of infinite total measure – the convex sequel

Let (Un)n be an arbitrary sequence of open convex subsets of the unit disk D(0,1)⊆R2 s.t. ∑∞n=0λ(Un)=∞ (where λ is the Lebesgue measure). Does there exist a sequence (qn)n in R2 s.t. D(0,1)⊆⋃∞n=0(qn+Un)? With the notation qn+Un, I mean qn+Un:={x∈R2|x−qn∈Un} This question is very similar to this one, but I was encouraged in the comments … Read more

Identifying poisoned wines, with a twist

(This is a joint musing with Andrew Gordon and Wyatt Mackey) There is a classic, elementary riddle, discussed before on MO and math.SE: suppose you have 1000 bottles of wine, and one is poisoned. The poison is slow-acting, and will not cause any negative effects until 24 hours after consumed. How many taste-testers (for the … Read more

(Non)existence of mirrors with more than two foci

Do there exist any mirrors M in d-dimensional Euclidean space Rd for which there exist three different points x1, x2, x3∈Rd such that if any ray of light passes through one of the points xi it automatically passes through the other two after a number of reflections on M? The question needs some formalisation for … Read more

Truncated Exponential Series Modulo pp: Deeper meaning for a Putnam Question.

Apparently B6 of the Putnam this year asked: Suppose p is an odd prime. Prove that for n∈{0,1,2…p−1}, at least p+12 of the numbers ∑p−1k=0k!nk are not divisble by p. With some rearrangements, this is equivalent to showing that Ep(z):=p−1∑k=0zkk! has at most p−12 zeros. A proof of this is at the end. My question … Read more