in Wildfires on Logistic regression, Geospatial data, Climate
Part 1 of predicting wildfire incidence. This goes over preprocessing all the input data (wildfires, land usage, and climatology).
These series of blog posts will go over our methods for predicting wildfires across the United States given historical wildfire incidences, land usage data, and high-resolution climatology data. Please refer to the Wildfire Incidence Prediction (CU) to learn more about an overview of the project, including the presentation and the submitted final report.
Consider the isosceles triangle with base length, $b=16$, and legs, $L=17$.
By using the Pythagorean theorem it can be seen that the height of the triangle, $h=\sqrt{17^2-8^2}=15$, which is one less than the base length.
With $b=272$ and $L=305$, we get $h=273$, which is one more than the base length, and this is the second smallest isosceles triangle with the property that $h=b\pm1$.
Find $\sum L$ for the twelve smallest isosceles triangles for which $h=b\pm1$ and $b,L$ are positive integers.
We shall define a square lamina to be a square outline with a square “hole” so that the shape possesses vertical and horizontal symmetry. For example, using exactly thiry-two square tiles we can form two different square lamina.
With one-hundred tiles, and not necessarily using all of the tiles at one time, it is possible to form forty-one different square laminae.
Using up to one million tiles how many different square laminae can be formed?
In the hexadecimal number system numbers are represented using 16 different digits: \[0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F\]
The hexadecimal number AF when written in the decimal number system equals $10\times 16+15=175$.
In the 3-digit hexadecimal numbers 10A, 1A0, A10, and A01 the digits 0, 1, and A are all present. Like numbers written in base ten we write hexadecimal numbers without leading zeroes.
How many decimal numbers containing at most sixteen hexadecimal digits exist with all of the digits 0, 1, and A present at least once? Give your answer as a hexadecimal number.
(A,B,C,D,E and F in upper case, without any leading or trailing code that marks the number as hexadecimal and without leading zeroes, e.g. 1A3F and not: 1a3f and not 0x1a3f and not $1A3F and not #1A3F and not 0000001A3F)
In the following equation, $x$, $y$, and $n$ are positive integers. \[\frac{1}{x} + \frac{1}{y} = \frac{1}{n}\]
It can be verified that when $n=1260$ there are 113 distinct solutions and this is the least value of $n$ for which the total number of distinct solutions exceeds one hundred.
What is the least value of $n$ for which the number of distinct solutions exceeds four million?
The smallest number expressible as the sum of a prime square, prime cube, and prime fourth is 28. In fact, there are exactly four numbers below fifty that can be expressed in such a way: \[28 = 2^2+2^3+2^4 \\ 33 = 3^2+2^3+2^4 \\ 49 = 5^2+2^3+2^4 \\ 47 = 2^2+3^3+2^4\]
How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?
Euler’s Totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, $\phi(9)=6$. The number 1 is considered to be relatively prime to every positive number, so $\phi(1)=1$.
Interestingly, $\phi(87109)=79180$, and it can be seen that $87109$ is a permutation of $79180$.
Find the value of $n, 1<n<10^7$, for which $\phi(n)$ is a permutation of $n$ and the ratio $n/\phi(n)$ produces a minimum.
Consider the following “magic” 3-gon ring, filled with 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 the set: 4,3,2; 6,2,1; 5,1,3.
It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
Total
Solution Set
9
4,2,3; 5,3,1; 6,1,2
9
4,3,2; 6,2,1; 5,1,3
10
2,3,5; 4,5,1; 6,1,3
10
2,5,3; 6,3,1; 4,1,5
11
1,4,6; 3,6,2; 5,2,4
11
1,6,4; 5,4,2; 3,2,6
12
1,5,6; 2,6,4; 3,4,5
12
1,6,5; 3,5,4; 2,4,6
By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.
Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string for a “magic” 5-gon ring?
