1617-257/Riddle Repository

Riddle Repository

A collection of the riddles posed at the beginning of each lecture

Date	Riddle	Solutions, Discussion, etc...
Sept 12	We want to compute $(x^{x})'$. Prof. A claims $(x^{n})'=nx^{n-1}$, so $(x^{x})'=xx^{x-1}=x^{x}$ Prof. B claims $(a^{x})'=a^{x}\log(a)$, so $(x^{x})'=x^{x}\log(x)$ Smart student says $(x^{x})'=x^{x}+x^{x}\log(x)$. Why is the derivative the sum of the Prof's solutions?
Sept 14	Can all of $\mathbb {R} ^{2}$ be covered by a set of disjoint, non-degenerate, circles? What about $\mathbb {R} ^{3}$? $\mathbb {R} ^{4}$?
Sept 16	Can you find uncountably many disjoint subsets of $\mathbb {R}$?
Sept 19	Can uncountably many Y shapes be fit into $\mathbb {R} ^{2}$?
Sept 21	On any pair of potatoes, can you draw a pair of 3D congruent curves? Hint (Hover)
Sept 23	Can you find uncountably many subsets of $\mathbb {N}$ s.t. the intersection of any two of them is finite?
Sept 26	In how many ways can you place 4 different points in the Euclidean plane, such that there are 2 different distances between all the points?
Sept 28	Can you find uncountably many subsets of $\mathbb {N}$ s.t. for any two of them A and B, (A $\subset$ B) or (B $\subset$ A)?
Sept 30	In a random 13-element subset of 1,2,...,52, what is the average value of the smallest element? (Credit: Yujia Yin)
Oct 3	A spherical loaf of bread is put in a bread cutting machine. Which slice gets the most crust?
Oct 5	Can you write the function $f(x,y)=1+xy+x^{2}y^{2}$ as $f(x,y)=\sum _{k=1}^{2}g_{k}(x)h_{k}(x)$? (Credit: Yujia Yin)
Oct 7	Prove: If you tile a rectangle (whose sides are not integers) with rectangles, at least one of those will have both sides non-integer.
Oct 10	(Thanksgiving holiday - University closed)
Oct 12	Players A and B alternate placing 1 x 2, 1 x 3, and 1 x 4 lego pieces (as they choose) on a 19 x 21 board, with no layering and no overlaps. If you cannot place a piece, you lose. Who would you rather be A or B? What if the overall size was 20 x 20?
Oct 14
Oct 17
Oct 19