Articles by xi'an

puzzles & riddles

January 2, 2021 | xi'an

A rather simplistic game on the Riddler of 18 December: …two players, each of whom starts with a whole number of points. Players take turns “attacking” each other, which involves subtracting their own number of points from their opponent’s until one of the players is out of points. Easy to ...
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how many Friday 13th?

December 17, 2020 | xi'an

A short Riddler’s riddle on the maximum number of Fridays 13th over a calendar year, of which I found 9 by a dumb exploration : bi=c(1:31,1:29,1:31,1:30,1:31,1:30,1:31,1:31,1:30,1:31,1:30,1:31) oy=bi[-60] for(j in 0:(length(cy
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around the table

December 1, 2020 | xi'an

The Riddler has a variant on the classical (discrete) random walk around a circle where every state (but the starting point) has the same probability 1/(n-1) to be visited last. Surprising result that stems almost immediately from the property that, leaving from 0, state a is visited couterclockwise before state b__...
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Bernoulli factory in the Riddler

November 30, 2020 | xi'an

“Mathematician John von Neumann is credited with figuring out how to take a p biased coin and “simulate” a fair coin. Simply flip the coin twice. If it comes up heads both times or tails both times, then flip it twice again. Eventually, you’ll get two different flips — either ...
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the riddle(r) of the certain winner losing in the end

November 24, 2020 | xi'an

Considering a binary random walk, starting at zero, what is the probability of being almost sure of winning at some point only to lose at the end? This is the question set by the post-election Riddler, with almost sure meaning above 99% and the time horizon set to n=101 steps (it ...
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Le Monde puzzle [#1164]

November 15, 2020 | xi'an

The weekly puzzle from Le Monde is quite similar to older Diophantine episodes (I find myself impossible to point out): Give the maximum integer that cannot be written as 105x+30y+14z. Same question for 105x+70y+42z+30w. These are indeed Diophantine equations and the existence of a solution ...
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on arithmetic derivations of square roots

November 12, 2020 | xi'an

An intriguing question made a short-lived appearance on the CodeGolf section of Stack Exchange, before being removed, namely the (most concise possible) coding of an arithmetic derivation of the square root of an integer, S, with a 30 digit precision and using only arithmetic operators. I was not aware of the ...
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Le Monde puzzle [#1158]

November 9, 2020 | xi'an

A weekly puzzle from Le Monde on umbrella sharing: Four friends, Antsa, Cyprien, Domoina and Fy, are leaving school to return to their common housing. It is raining and they only have one umbrella with only room for two. Given walking times, x¹, x², x³ and x⁴, find the fastest ...
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asymmetric information

November 3, 2020 | xi'an

The Riddler of 16 October had the following puzzle: Take a real number θ uniformly distributed over (0,100). Among three players, the winner is whoever guessed the closest price without going over θ. In the event all guesses exceeded θ, the contestant with the lowest (and therefore closest) guess is declared the winner. The second […]
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sampling w/o replacement except when replacing

November 2, 2020 | xi'an

Another Riddle(r), considering a box with M myrtle balls and D dandelion balls. Drawing balls without replacement while they stay of the same color as the initial draw, else put back the last ball and repeat the process until all balls are drawn. The funny thing is that, unless ...
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inverse Gaussian trick [or treat?]

October 28, 2020 | xi'an

When preparing my mid-term exam for my undergrad mathematical statistics course, I wanted to use the inverse Gaussian distribution IG(μ,λ) as an example of exponential family and include a random generator question. As shown above by a Fortran computer code from Michael, Schucany and Haas, a simple version can be ...
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artificial EM

October 27, 2020 | xi'an

When addressing an X validated question on the use of the EM algorithm when estimating a Normal mean, my first comment was that it was inappropriate since there is no missing data structure to anchor by (right preposition?). However I then reflected upon the infinite number of ways to demarginalise ... [Read more...]

parking riddle

October 22, 2020 | xi'an

The Riddler of this week had a quick riddle: if one does want to avoid parallel parking a car over a six spot street, either the first spot is available or two consecutive spots are free. What is the probability this happens with 4 other cars already parked (at random)? While ...
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Fermat’s Riddle

October 15, 2020 | xi'an

·A Fermat-like riddle from the Riddler (with enough room to code on the margin) An  arbitrary positive integer N is to be written as a difference of two distinct positive integers. What are the impossible cases and else can you provide a list of all distinct representations? Since the problem ...
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understanding elections through statistics [book review]

October 11, 2020 | xi'an

A book to read most urgently if hoping to take an informed decision by 03 November! Written by a political scientist cum statistician, Ole Forsberg. (If you were thinking of another political scientist cum statistician, he wrote red state blue state a while ago! And is currently forecasting the outcome of ...
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Le Monde puzzle [#1159]

October 5, 2020 | xi'an

The weekly puzzle from Le Monde is quite similar to #1157: Is it possible to break the ten first integers, 1,…,10, into two groups such that the sum over the first group is equal to the product over the second? Is it possible that the second group is of cardinal 4? of cardinal 3? ...
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Le Monde puzzle [#1157]

September 30, 2020 | xi'an

The weekly puzzle from Le Monde is an empty (?) challenge: Kimmernaq and Aputsiaq play a game where Kimmernaq picks ten different integers between 1 and 100, and Aputsiaq must find a partition of these integers into two groups with identical sums. Who is winning? Indeed, if the sums are equal, then the ...
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Le Monde puzzle [#1155]

September 25, 2020 | xi'an

The weekly puzzle from Le Monde is another Sudoku challenge: Anahera and Wiremu play a game for T rounds. They successively pick a digit between 1 and 3, never repeating the previous one, and sum these digits. The last to play wins if the sum is a multiple of 3. Who is the ...
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the limits of R

August 10, 2020 | xi'an

It has been repeated many times on many platforms, the R (or R⁰) number is not a great summary about the COVID-19 pandemic, see eg Rossman’s warning in The Conversation, but Nature chose to stress it one more time (in its 16 Jul edition). Or twice when considering a similar ...
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