Dyalog APL Problem Solving Competition 2021

https://www.dyalog.com/news/147/420/Meet-The-2021-APL-Problem-Solving-Competition-Winners.htm

Overview

The competition consists of two parts:

• Phase 1 asks you to solve 10 puzzles by writing short APL functions, allowing you to demonstrate array-oriented thinking. You can begin without registering — your solutions should be stored by your browser until you decide to register and submit them (not all browsers do this).
• Phase 2 comprises a collection of more difficult problems, each having one or two tasks. In addition to requiring array-oriented thinking, this enables you to show off your ability to write larger amounts of well-documented, high-quality, code.

The problem specifications given on this site can also be downloaded as PDF files:

PDF of the Phase 1 problem set

PDF of the Phase 2 problem set

Approach

APL lets you do most kinds iteration and reduction with symbolic operators, and compose these into dense functions with "glue" operators. There are three syntaxes to choose from for a function definition, which are called a tradfn, dfn, or tacit function. I used only dfn and tacit functions in my programs because they compose the most elegantly, together with TryAPL as a REPL to come up with answers one piece at a time. For example, to make a tacit function that checks if its input is a magic square:

With any nxn array,

      3 3⍴4 9 2 3 5 7 8 1 6
4 9 2
3 5 7
8 1 6

we can take its reverse and transpose,

      (⊂∘⍉,⊂∘⌽) 3 3⍴4 9 2 3 5 7 8 1 6
┌─────┬─────┐
│4 3 8│2 9 4│
│9 5 1│7 5 3│
│2 7 6│6 1 8│
└─────┴─────┘

concatenate their diagonals,

      ((⊢, 1 1∘⍉)¨(⊂∘⍉,⊂∘⌽)) 3 3⍴4 9 2 3 5 7 8 1 6
┌───────┬───────┐
│4 3 8 4│2 9 4 2│
│9 5 1 5│7 5 3 5│
│2 7 6 6│6 1 8 8│
└───────┴───────┘

remove their nesting,

      ((⊃,/)∘((⊢,1 1∘⍉)¨(⊂∘⍉,⊂∘⌽))) 3 3⍴4 9 2 3 5 7 8 1 6
4 3 8 4 2 9 4 2
9 5 1 5 7 5 3 5
2 7 6 6 6 1 8 8

sum their columns,

      ((+⌿)∘(⊃,/)∘((⊢,1 1∘⍉)¨(⊂∘⍉,⊂∘⌽))) 3 3⍴4 9 2 3 5 7 8 1 6
15 15 15 15 15 15 15 15

and check their equality,

      ((∧/⊃=⊢)∘(+⌿)∘(⊃,/)∘((⊢,1 1∘⍉)¨(⊂∘⍉,⊂∘⌽))) 3 3⍴4 9 2 3 5 7 8 1 6
1

which makes our final program ((∧/⊃=⊢)∘(+⌿)∘(⊃,/)∘((⊢,1 1∘⍉)¨(⊂∘⍉,⊂∘⌽))). That string alone still looks arcane to me, but broken down piece by piece it becomes much clearer, and remains shorter winded than most other languages.

Result: Getting there!

I placed third in Phase II and took home bronze for the states, and I am looking forward to using what I learned to do better next year.