Publications by xi'an
Le Monde puzzle [#1141]
The weekly puzzle from Le Monde is in honour of John Conway, who just passed away, ending up his own game of life: On an 8×8 checker-board, Alice picks n squares as “infected”. She then propagates the disease by having each square with least two infected neighbours to become infected as well. What is the minimal value of n for the entire bo...
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more games of life
Another puzzle in memoriam of John Conway in The Guardian: Find the ten digit number, abcdefghij. Each of the digits is different, and a is divisible by 1 ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible b...
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another Bernoulli factory
A question that came out on X validated is asking for help in figuring out the UMVUE (uniformly minimal variance unbiased estimator) of (1-θ)½ when observing iid Bernoulli B(θ). As it happens, there is no unbiased estimator of this quantity and hence not UMVUE. But there exists a Bernoulli factory producing a coin with probability (1-θ)½ fro...
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not a Bernoulli factory
A Riddler riddle I possibly misunderstood: Four isolated persons are given four fair coins, which can be either flipped once or returned without being flipped. If all flipped coins come up heads, the team wins! Else, if any comes up tails, or if no flip at all is done, it looses. Each person is further given an independent U(0,1) realisation. Wh...
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a new Monty Hall riddle
The Riddler was sort of feeling the rising boredom of being under lockdown when proposing the following variant to the Monty Hall puzzle: There are zero to three goats, with a probability ¼ each, and they are allocated to different doors uniformly among the three doors of the show. After the player chooses a door, Monty opens another door hiddi...
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minimax, maximin or plain
A simple riddle from The Riddler on choosing between the maximum between two minima of two throws of an N-face dice, the minimum between two maxima of two throws of an N-face dice, and a single throw. Since maximin is always less than maximin, second choice is always worse than first and a stochastic domination version of the argument shows the s...
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Le Monde puzzle [#1146]
The weekly puzzle from Le Monde is once more disappointing. Everyday of the month, take 0, 1 or 2 units. If one unit taken past day, next day none can be taken. If two units taken two day ago, none can be taken the current day. What is the strategy maximising the number of units for February? March? April? Generalise to 0,..,3 units taken each...
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the strange occurrence of the one bump
When answering an X validated question on running an accept-reject algorithm for the Gamma distribution by using a mixture of Beta and drifted (bt 1) Exponential distributions, I came across the above glitch in the fit of my 10⁷ simulated sample to the target, apparently displaying a wrong proportion of simulations above (or below) one. a=.9 g...
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Le Monde puzzle [#1147]
The weekly puzzle from Le Monde is not much, again: A number A is such that (1) the difference between two digits is never 1 (2) two digits are equal to 2 (3) it only involves 3 different digits (4) it is a multiple of 4 (5) the sum of two of the digits is 5 (6) the product of the digits is not a multiple of 6 (7) the digits are between 0 and ...
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overlap, overstreched
An interesting challenge on The Riddler on the probability to see a random interval X’ing with all other random intervals when generating n intervals from Dirichlet D(1,1,1). As it happens the probability is always 2/3, whatever n>1, as shown by the R code below (where replicate cannot be replaced by rep!): qro=function(n,T=1e3){ quo=functio...
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