Publications by xi'an
preserving frequencies without resampling
An interesting question came up on X validated a few days ago: given a probability vector p=(p¹,…,p⁷), is there a way to pick 5 values in {1,…,7} without replacement and still preserve the probability repartition in the resulting sample? In other words, is there a sampling without replacement strategy that leads to for i=1,…,7..? Unless...
2170 sym R (207 sym/1 pcs) 10 img
Le Monde puzzle [#952]
A quite simple Le Monde mathematical puzzle again with Alice and Bob: In a multiple choice questionnaire with 50 questions, Alice gets a score s such that Bob can guess how many correct (+5 points), incorrect (-1 point) and missing (0 point) Alice got when adding that Alice could not have gotten s-2 or s+2. What is Alice’s score? A first poin...
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Le Monde puzzle [#954]
A square Le Monde mathematical puzzle: Given a triplet (a,b,c) of integers, with a? Can you find the triplet (a,b,c) that produces the sum a+b+c closest to 1000? This is a rather interesting challenge and a brute force resolution does not produce interesting results. For instance, using the function is.whole from the package Rpmfr, the R functi...
1280 sym R (772 sym/4 pcs) 6 img
another riddle
A very nice puzzle on The Riddler last week that kept me busy on train and plane rides, runs and even in between over the weekend. The core of the puzzle is about finding the optimal procedure to select k guesses about the value of a uniformly random integer x in {a,a+1,…,b}, given that each guess y produces the position of x respective to y (l...
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Le Monde puzzle [#956]
A Le Monde mathematical puzzle with little need of R programming: Does there exist a function f from N to N such that (i) f is (strictly) increasing, (ii) f(n)≥n, and (iii) f²(n)=f(f(n))=3n? Indeed, the constraints imply (i) f²(0)=0, hence that that f(0)=0, (ii) f(1)=2 as it can be neither 1 (else f²(1) would be equal to 1, not to 3) nor ...
1957 sym R (302 sym/1 pcs) 8 img
Statistical rethinking [book review]
Statistical Rethinking: A Bayesian Course with Examples in R and Stan is a new book by Richard McElreath that CRC Press sent me for review in CHANCE. While the book was already discussed on Andrew’s blog three months ago, and [rightly so!] enthusiastically recommended by Rasmus Bååth on Amazon, here are the reasons why I am quite impressed by...
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Le Monde puzzle [#958]
A knapsack Le Monde mathematical puzzle: Given n packages weighting each at most 5.8kg for a total weight of 300kg, is it always possible to allocate these packages to 12 separate boxes weighting at most 30kg each? weighting at most 29kg each? This can be checked by brute force using the following R code #generate packages paca=runif(1,0,5.8)...
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Le Monde puzzle [#959]
Another of those arithmetic Le Monde mathematical puzzle: Find an integer A such that A is the sum of the squares of its four smallest dividers (including1) and an integer B such that B is the sum of the third poser of its four smallest factors. Are there such integers for higher powers? This begs for a brute force resolution checking the integ...
1817 sym R (270 sym/2 pcs) 8 img
an integer programming riddle
A puzzle on The Riddler this week that ends up as a standard integer programming problem. Removing the little story around the question, it boils down to optimise 200a+100b+50c+25d under the constraints 400a+400b+150c+50d≤1000, b≤a, a≤1, c≤8, d≤4, and (a,b,c,d) all non-negative integers. My first attempt was a brute force R code since t...
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Le Monde puzzle [#960]
An arithmetic Le Monde mathematical puzzle: Given an integer k>1, consider the sequence defined by F(1)=1+1 mod k, F²(1)=F(1)+2 mod k, F³(1)=F²(1)+3 mod k, &tc. [With this notation, F is not necessarily a function.] For which value of k is the sequence the entire {0,1,…,k-1} set? This leads to an easy brute force resolution, for instance w...
2106 sym R (168 sym/2 pcs) 6 img