The second Le Monde mathematical puzzle in the new competition is sheer trigonometry: When in the above figures both triangles ABC are isosceles and the brown segments are all of length 25cm, find the angle in A and the value of DC², respectively. This could have been solved by R ...

Recalling Le Monde mathematical puzzle  first competition problem For the X table below, what are the minimal number of lights that are on (green) to reach the minimal and maximal possible numbers of entries (P) with an even (P as pair) number of neighbours with lights on? In the illustration ...

A second Riddle(r), with a puzzle related with the integer set Ð={,12,3,…,N}, in that it summarises as Given a random walk on Ð, starting at the middle N/2, with both end states being absorbing states, and a uniform random move left or right of the current value to the (integer) ...

A penultimate Le Monde mathematical puzzle  before the new competition starts [again!] For a game opposing 40 players over 12 questions, anyone answering correctly a question gets as reward the number of people who failed to answer. Alice is the single winner: what is her minimal score? In another round, Bob is ...

There will be a symposium on ABC in Montréal this coming December, the day before NIPS, in a continuation of past years NIPS workshops. While invited speakers and panelists have been selected by the committee, a call for papers is open. Note that in continuation with the best “ABC ...

I made the “capital” mistake of walking across the entire NUS campus this morning, which is quite green and pretty, but which almost enjoys an additional dimension brought by such an intense humidity that one feels having to get around this humidity!, a feature I have managed to completely erase ...

In the Riddler of August 18, two riddles connected with the integer set Ð={2,3,…,10}: Given a permutation of Ð, what is the probability that the most likely variation (up or down) occurs at each term? Given three players choosing three different integers in Ð sequentially, and rewards in Ð allocated to the closest of the […]

I came across this question on X validated that wondered about one of our examples in Monte Carlo Statistical Methods. We have included a section on Laplace approximations in the Monte Carlo integration chapter, with a bit of reluctance on my side as this type of integral approximation does not ...

A simple (summertime?!) arithmetic Le Monde mathematical puzzle A “powerful integer” is such that all its prime divisors are at least with multiplicity 2. Are there two powerful integers in a row, i.e. such that both n and n+1 are powerful?  Are there odd integers n such that n² – 1 is ...

A simple Le Monde mathematical puzzle none too geometric: Find square triangles which sides are all integers and which surface is its perimeter. Extend to non-square rectangles. No visible difficulty by virtue of Pythagore’s formula: for (a in 1:1e4) for (b in a:1e4) if (a*b==2*(a+b+...

A griddy Le Monde mathematical puzzle: On a 4×5 regular grid, find how many nodes need to be turned on to see all 3×4 squares to have at least one active corner in case of one arbitrary node failing.  Repeat for a 7×9 grid. The question is open to simulated annealing, as in ...

“We demonstrate HMC’s sensitivity to these parameters by sampling from a bivariate Gaussian with correlation coefficient 0.99. We consider three settings (ε,L) = {(0.16; 40); (0.16; 50); (0.15; 50)}” Ziyu Wang, Shakir Mohamed, and Nando De Freitas. 2013 In an experiment with my PhD student Changye Wu (who wrote all R codes used below), we […]

The most recent riddle on the Riddler consists in finding the shorter sequence of digits (in 0,1,..,9) such that all 10⁴ numbers between 0 (or 0000) and 9,999 can be found as a group of consecutive four digits. This sequence is obviously longer than 10⁴+3, but how long? On my trip to Brittany last weekend, […]

Once in a while, or a wee bit more frequently (!), it proves impossible to communicate with a contributor of a question on X validated. A recent instance was about simulating from a multivariate kernel density estimate where the kernel terms at x¹,x²,… are Gaussian kernels applied to the inverses ...

As an illustration of the lack of convergence of the Gibbs sampler applied to the two “conditionals” defined in the Bayesian GANs paper discussed yesterday, I took the simplest possible example of a Normal mean generative model (one parameter) with a logistic discriminator (one parameter) and implemented the scheme (during ...

An easy arithmetic Le Monde mathematical puzzle again: If coins come in units of 1, x, and y, what is the optimal value of (x,y) that minimises the number of coins representing an arbitrary price between 1 and 149?  If the number of units is now four, what is the optimal choice? ...

A quick riddler resolution during a committee meeting (!) of a short riddle: 36 houses stand in a row and collapse at times t=1,2,..,36. In addition, once a house collapses, the neighbours if still standing collapse at the next time unit. What are the shortest and longest lifespans of this row? Since ...

The proposed solution of the riddle from the Riddler discussed here a few weeks ago is rather approximative, in that the distribution of when the n-sample is made of iid Normal variates is (a) replaced with the distribution of one arbitrary minimum and (b) the distribution of the minimum is ...

A combinatoric Le Monde mathematical puzzle of limited size: When the only allowed move is to switch two balls from adjacent boxes, what is the minimal number of moves to return all balls in the above picture to their respective boxes? Same question with six boxes and 12 balls. The question ...