1617-257/Riddle Repository

Riddle Repository

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

Date of Lecture	Riddle	Solutions, Discussion, etc...
Sept 12	We want to compute [math]\displaystyle{ (x^x)' }[/math]. Prof. A claims [math]\displaystyle{ (x^n)'=nx^{n-1} }[/math], so [math]\displaystyle{ (x^x)' = xx^{x-1} = x^x }[/math] Prof. B claims [math]\displaystyle{ (a^x)' = a^x\log(a) }[/math], so [math]\displaystyle{ (x^x)' = x^x\log(x) }[/math] Smart student says [math]\displaystyle{ (x^x)' = x^x + x^x\log(x) }[/math]. Why is the derivative the sum of the Prof's solutions?
Sept 14	Can all of [math]\displaystyle{ \mathbb{R}^2 }[/math] be covered by a set of disjoint, non-degenerate, circles? What about [math]\displaystyle{ \mathbb{R}^3 }[/math]? [math]\displaystyle{ \mathbb{R}^4 }[/math]?
Sept 16	Can you find uncountably many disjoint subsets of [math]\displaystyle{ \mathbb{R} }[/math]?
Sept 19	Can uncountably many Y shapes be fit into [math]\displaystyle{ \mathbb{R}^2 }[/math]?
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 [math]\displaystyle{ \mathbb{N} }[/math] 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 [math]\displaystyle{ \mathbb{N} }[/math] s.t. for any two of them A and B, (A [math]\displaystyle{ \subset }[/math] B) or (B [math]\displaystyle{ \subset }[/math] 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 [math]\displaystyle{ f(x, y) = 1 + xy + x^2y^2 }[/math] as [math]\displaystyle{ f(x, y) =\sum_{k=1}^{2}g_k(x) h_k(x) }[/math]? (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	No riddle. Discusses past riddle from Sept 19.
Oct 17	An ant walks at 1cm/sec along a super-rubber-band that stretches at 1m/sec. Will it ever reach the other end? Why not? How long will it take? (Credit: Kodiak Jackson)
Oct 19	How far can you go with n Jenga blocks?
Oct 21	Can you place 6 identical real-life Jenga blocks/chalks such that any two of the will touch each other?
Oct 24	Can you pack 125 1 x 2 x 4 boxes inside a 10 x 10 x 10 cube?
Oct 26	Can you pack 21 3 x 1 rectangles on an 8 x 8 board? Any constraints on where the missing piece would be?
Oct 28	No riddle.
Oct 31	A total of k kids share a loot of n indivisible candies. The first proposes a split. If not accepted by a strict majority, she leaves and the second proposes a split... etc. How will the loot be split?
Nov 2	Abhishek is at the centre of a circle of radius 100m. On the circle is a Lion. [math]\displaystyle{ V_L }[/math] = [math]\displaystyle{ 4V_A }[/math]. Help save Abhishek, by giving him a strategy that can always get him out of the circle, given that the Lion is very intelligent.
Nov 4	Dmitry is at the centre of a stadium of radius 100m without any exit. A Lion is also in the circle. [math]\displaystyle{ V_D }[/math] = [math]\displaystyle{ V_A }[/math]. Given that Dmitry and the Lion are very intelligent, how long can Dmitry survive?
Nov 7	(Fall reading break - No classes)
Nov 9	Can you find two irrational numbers x and y such that [math]\displaystyle{ x^y }[/math] is a rational number?
Nov 14	Can you find a continuous f: [0,1] -> [0,1] with f(0) = 0 and f(1) = 1 which is differentiable with Df = 0 except on a set of measure 0?
Nov 16	Can you cover a diameter 100 disk with 99 (possibly overlapping) 100 x 1 bands?