解 Perl Weekly Challenge 089 -- 公因數之和與魔方

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Perl Weekly Challenge 089 兩題都是數學謎題呢。姑且就先用暴力法來解吧。

TASK #1 › GCD Sum

Submitted by: Mohammad S Anwar

You are given a positive integer $N.

Write a script to sum GCD of all possible unique pairs between 1 and $N.

Example 1:

Input: 3 Output: 3

gcd(1,2) + gcd(1,3) + gcd(2,3)

Example 2:

Input: 4 Output: 7

gcd(1,2) + gcd(1,3) + gcd(1,4) + gcd(2,3) + gcd(2,4) + gcd(3,4)


如果直接將題意「翻譯」成 Raku 程式碼的話,就會得到直觀暴力解:

(1..$N).combinations(2).map(-> ($a, $b) { $a gcd $b }).sum()

TASK #2 › Magical Matrix

Submitted by: Mohammad S Anwar

Write a script to display matrix as below with numbers 1 - 9. Please make sure numbers are used once.

[ a b c ]
[ d e f ]
[ g h i ]

So that it satisfies the following:

a + b + c = 15
d + e + f = 15
g + h + i = 15
a + d + g = 15
b + e + h = 15
c + f + i = 15
a + e + i = 15
c + e + g = 15

這題就是用程式解 3×3 魔方啦。如果完全不事先分析魔方特性,只靠暴力法搜尋來解的話,就等同於先產生出 1..9 的排列,再過濾掉任何不符前述八條算式者。

(1..9).permutations.grep(
    -> ($a, $b, $c, $d, $e, $f, $g, $h, $i) {
        all(
            $a + $b + $c == 15,
            $d + $e + $f == 15,
            $g + $h + $i == 15,
            $a + $d + $g == 15,
            $b + $e + $h == 15,
            $c + $f + $i == 15,
            $a + $e + $i == 15,
            $c + $e + $g == 15,
        )
    }
)

.permutations 函式能逐一產生出所有排列。因為這裡是讓他產生九個數字的排列, 所以後方接的 grep 就有九個參數,這麼一來裡面的判斷條件就可以直接從題目本文照抄過來。