# Blog Archives

## gap frequencies [& e]

April 28, 2016
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A riddle from The Riddler where brute-force simulation does not pay: For a given integer N, pick at random without replacement integers between 1 and N by prohibiting consecutive integers until all possible entries are exhausted. What is the frequency of selected integers as N grows to infinity? A simple implementation of the random experiment

## Le Monde puzzle [#960]

April 27, 2016
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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. For which value of k is the sequence the entire {0,1,…,k-1} set? This leads to an easy brute force resolution, for

## an integer programming riddle

April 20, 2016
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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 there are only

## Le Monde puzzle [#959]

April 19, 2016
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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

## Le Monde puzzle [#958]

April 10, 2016
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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 and

## Statistical rethinking [book review]

April 5, 2016
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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 enthusiastically recommended by Rasmus Bååth on Amazon, here are the reasons why I

## Le Monde puzzle [#956]

April 4, 2016
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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

## another riddle

March 28, 2016
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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

## Le Monde puzzle [#954]

March 24, 2016
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A square Le Monde mathematical puzzle: Given a triplet (a,b,c) of integers, with a<b<c, it satisfies the S property when a+b, a+c, b+c, a+b+c are perfect squares such that a+c, b+c, and a+b+c are consecutive squares. For a given a, is it always possible to find a pair (b,c) such (a,b,c) satisfies S? Can you

## Le Monde puzzle [#952]

March 18, 2016
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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.