# RogerBW's Blog

 Perl Weekly Challenge 89: magic sum 05 December 2020 I’ve been doing the Perl Weekly Challenges. The latest involved GCDs and magic squares. (Note that this is open until 6 December 2020.) TASK #1 › GCD Sum You are given a positive integer `\$N`. Write a script to sum GCD of all possible unique pairs between `1` and `\$N`. There doesn't appear to be an obvious optimisation listed at OEIS, so I did this naïvely. ``````sub gs { my \$n=shift; my \$tot=\$n-1; # gcd(1,2)+gcd(1,3)+...+gcd(1,n) foreach my \$a (2..\$n-1) { \$tot+=sum(map {gcd(\$a,\$_)} (\$a+1..\$n)); } return \$tot; } `````` Perl doesn't have a built-in `gcd`, so rather than use one from CPAN I implemented Stein's algorithm (on the basis that I've written Euclid's before). This is rather more complex than the main problem, but I made it non-recursive for clarity.. ``````sub gcd { my (\$a,\$b)=@_; if (\$a==0) { return \$b; } if (\$b==0) { return \$a; } my \$d=0; while (1) { if (\$a == \$b) { return \$a << \$d; } my \$aa=(\$a % 2 == 0); my \$bb=(\$b % 2 == 0); `````` We do different things based on whether they're both even, only one is, or neither is. `````` if (\$aa && \$bb) { \$a >>= 1; \$b >>= 1; \$d++; } elsif (\$aa) { \$a >>= 1; } elsif (\$bb) { \$b >>= 1; } else { my \$c=abs(\$a-\$b); \$a=min(\$a,\$b); \$b=\$c >> 1; } } } `````` Raku came out basically the same, except that `sum` and `min` are built-in and its binary shift operators look weird. Python has its own `gcd`, as does Ruby, and they basically look the same too. But Rust… all right, I had to write a `min` function too. (Or steal it out of the Blandy/Orendorff Programming Rust.) TASK #2 › Magical Matrix Write a script to display matrix as below with numbers 1 - 9. Please make sure numbers are used once. i.e. a 3×3 magic square. But there is only one 3×3 magic square (modulo reflections and rotations) and that would have been boring. So instead I used the Siamese algorithm (de la Loubère, 1688), with a minor variation so that indices were always incremented (so as to avoid, when I got to the Rust version, the conversions I had to do last time). This should produce a magic square of any odd-numbered size, with any starting number and increment. ``````sub ms { my (\$order,\$start,\$inc)=@_; my \$m; foreach (1..\$order) { push @{\$m},[(0) x \$order]; } my (\$x,\$y)=(int(\$order/2)+1,int(\$order/2)); my \$n=\$start; while (1) { \$m->[\$x][\$y]=\$n; \$n+=\$inc; my (\$xa,\$ya)=((\$x+1) % \$order,(\$y+1) % \$order); if (\$m->[\$xa][\$ya]>0) { (\$xa,\$ya)=((\$x+2) % \$order,\$y); if (\$m->[\$xa][\$ya]>0) { last; } } (\$x,\$y)=(\$xa,\$ya); } return \$m; } `````` Actually printing it out is a job for library code. Raku, Python and Ruby were similar, and even Rust is recognisably the same algorithm: ``````fn ms(order: usize, start: i64, inc: i64) -> Vec> { let mut m=vec![vec![0; order]; order]; let mut x: usize=(order/2)+1; let mut y: usize=order/2; let mut n=start; loop { m[x][y]=n; n += inc; let mut xa=(x+1) % order; let mut ya=(y+1) % order; if m[xa][ya] > 0 { xa=(x+2) % order; ya=y; if m[xa][ya] > 0 { break; } } x=xa; y=ya; } return m; } `````` Full code on github. Comments on this post are now closed. If you have particular grounds for adding a late comment, comment on a more recent post quoting the URL of this one. 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