코딩 연습

The minimum number of cubes to cover every visible face on a cuboid measuring \(3 \times 2 \times 1\) is twenty-two. If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one hundred and eighteen cubes to cover every visible face. However, the first layer on a..

The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: \(6^2 + 7^2 + 8^2 +9^2 + 10^2 + 11^2 + 12^2\). There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is \(4164\). Note that \(1 = 0^2 + 1^2\) has not been included as this problem is concerned with the squares of..

The radical of \(n\), \({\rm rad}(n)\), is the product of the distinct prime factors of \(n\). For example, \(504 = 2^3 \times 3^2 \times 7\), so \({\rm rad}(504) = 2 \times 3 \times 7 =42\). If we calculate \({\rm rad}(n)\) for \(1 \le n \le 10\), then sort them on \({\rm rad}(n)\), and sorting on \(n\) if the radical values are equal, we get: Unsorted Sorted \(n\) \({\rm rad}(n)\) \(n\) \({\rm..

Let \(p_n\) be the \(n\)th prime : \(2,\; 3,\; 5,\; 7,\; 11,\; \cdots,\) and let \(r\) be the remainder when \((p_n -1)^n + (p_n +1)^n\) is divided by \(p_n ^2\). For example, when \(n=3, \; p_3 = 5\), and \(4^3 + 6^3 = 280 \equiv 5\) mod \(25\). The least value of \(n\) for which the remainder first exceeds \(10^9\) is \(7037\). Find the least value of \(n\) for which the remainder first exceed..

The most naive way of computing \(n^{15}\) requires fourteen multiplications: \[ n \times n \times \cdots \times n = n^{15}\] But using a "binary" method you can compute it is six multiplications: \[\begin{split} n \times n &= n^2 \\ n^2 \times n^2 &= n^4 \\ n^4 \times n^4 &= n^8 \\ n^8 \times n^4 &= n^{12} \\ n^{12} \times n^2 &= n^{14} \\ n^{14} \times n &= n^{15} \end{split}\]However it is ye..

A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its color is noted. After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random. The player pays £1 to play and wins if they have taken more blue discs than red discs at the end of the game. If the game is played for four turns, the probabili..

Let \(r\) be the remainder when \((a-1)^n +(a+1)^n\) is divided by \(a^2\). For example, if \(a=7\) and \(n=3\), then \(r=42: \; 6^3 +8^3 = 728 \equiv 42 \; \rm mod\; 49\). And as \(n\) varies, so too will \(r\), but for \(a=7\) it turns out that \(r_{\rm max} =42\). For \(3 \le a \le 1000\), find \(\sum r_{\rm max}\). \((a-1)^n + (a+1)^n\) 을 \(a^2\) 으로 나누었을 때의 나머지를 \(r\) 라고 하자. 예를 들어, \(a=7\) 이..

Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set {2, 5, 47, 89, 631}, all of the elements belonging to it are prime. How many distinct sets containing each of the digits one through nine exactly once contain only prime elements? 1 부터 9까지 모든 자연수들을 한 번씩만 사용하여 자연수들을 만든다면 여러 가지 가능성이 있다. 예를 들어, 집합 {2, ..

Using a combination of black square tiles and oblong tiles chosen from: red tiles measuring two units, green tiles measuring three units, and blue tiles measuring four units, it is possible to tile a row measuring five units in length in exactly fifteen different ways. How many ways can a row measuring fifty units in length be tiled? 길이 1인 정사각형 모양의 검은색 타일과 길이 2인 직사각형 모양의 빨간색 타일, 길이 3인 직사각형 모양의 초..

A row of five black square tiles is to have a number of its tiles replaced with colored oblong tiles chosen from red (length two), green (length three), or blue (length four). If red tiles are chosen there are exactly seven ways this can be done. If green tiles are chosen there are three ways. And of blue tiles are chosen there are two ways. Assuming that colors cannot be mixed there are 7+3+2=1..