Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by ... [Read more...]

I found wp-codebox could highlight R code two years ago. This plugin is based on GeSHi to highlight source code internally.
Now there are many ways to highlight R syntax in the website. Pretty-R provided by Inside-R is a popular tool in the community.
Read More: 633 Words Totally [Read more...]

The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.Find the smallest cube for which exactly five permutations of its digits are cube.
Read More: 372 Words Totally [Read more...]

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ...
Square P4,n=n2 1, 4, 9, 16, 25, ...
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The square root of 2 can be written as an infinite continued fraction.
\( \sqrt{2} = 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+?}}}} \) \sqrt{2} = 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+?}}}}
The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].
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Increasing quantitative data generated from transcriptomics and proteomics require integrative strategies for analysis. Here, we present an R package, clusterProfiler that automates the process of biological-term classification and the enrichment analysis of gene clusters. The analysis module and visualization module were combined into a reusable workflow. Currently, clusterProfiler supports three ... [Read more...]

The Chinese character Jiong (囧) is now widely used for expressing ideas or feelings such as annoyance, shock, embarrassment, awkwardness, scorn.
The function plot of y=1/(x^2-1) looks very similar of this symbol.
I use ggplot2 to draw the symbol of Jiong.
Read More: 359 Words Totally [Read more...]

The K-means algorithm is a method to automatically cluster similar data examples together.
The intuition behind K-means is an iterative procedure that starts by guessing the initial centroids, and then refines this guess by repeatedly assigning examples to their closest centroids and then recomputing the centroids based on the assignments.
... [Read more...]

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, ... [Read more...]

The first two consecutive numbers to have two distinct prime factors are:
14 = 2 × 7
Read More: 278 Words Totally [Read more...]

Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...
Read More: 472 Words Totally [Read more...]

We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 through 5 pandigital.
The product 7254 is unusual, as the identity, 39 × 186 = 7254, containing multiplicand, multiplier, and product is 1 through 9 pandigital.
Read More: 488 Words Totally [Read more...]

In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:
1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).
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Starting in the top left corner of a 2x2 grid, there are 6 routes (without backtracking) to the bottom right corner.
How many routes are there through a 20x20 grid?
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The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.
Let d1 be the 1st digit, d2 be the 2nd digit, and so on. In this way, we note ... [Read more...]

We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.
What is the largest n-digit pandigital prime that exists?
Read More: 288 Words Totally [Read more...]

Take the number 192 and multiply it by each of 1, 2, and 3:
192 × 1 = 192
Read More: 450 Words Totally [Read more...]

Now we move on to the second part of the Exercise 5.2, which requires to implement regularized logistic regression using Newton's Method.
Plot the data: [Read more...]

The first part of the Exercise 5.1 requires to implement a regularized version of linear regression.
Adding regularization parameter can prevent the problem of over-fitting when fitting a high-order polynomial.
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Exercise 4 required implementing Logistic Regression using Newton's Method.
The dataset in use is 80 students and their grades of 2 exams, 40 students were admitted to college and the other 40 students were not. We need to implement a binary classification model to estimates college admission based on the student's scores on these two ... [Read more...]

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