May 01, 2020
Wassily Kandinsky, Composition VIII, 1923
Every month, Jane Street Capital post a puzzle on their website. This was the puzzle for April 2020. [1]
I started to solve the puzzle by hand in order to understand the problem space better and to find patterns. Solving the triangular grid by hand is quite easy for lower value of N. Example for N=9.
Some observations:
A solution is only possible if the number of dots is multiple of 3. This reduced the problem space by a third.
Number of dots = (N + 1) * N / 2
Number of triads = dots / 3
= (N + 1) * N / 6
N dots triads
  
1 1 0.333333333
2 3 1
3 6 2
4 10 3.333333333 < infeasible, as the number of dots is not divisible by 3
5 15 5
6 21 7
7 28 9.333333333
8 36 12
9 45 15
10 55 18.33333333
11 66 22
.
.
37 703 234.3333333
38 741 247
39 780 260
At this point my thinking was to automate the manual (by hand) method by creating a program based on the depthfirst search (DFS) backtracking algorithm. Backtracking can be used to solve many well known problems such as suduko [6] and the eight queens puzzle.
A concern was that the runtime might be very long for high values of N. Worse case the runtime is polynomial  O(n^k), with k somewhere between 24. However with the known constraints such as the the sides and corners, it might not be so bad.
The program I created was called triad.exe
. More about the program a bit later.
Running triad.exe
N dots triads solve
   
1 1 0.3 
2 3 1.0 1
3 6 2.0 
4 10 3.3 
5 15 5.0 
6 21 7.0 
7 28 9.3 
8 36 12.0 
9 45 15.0 1 < solve = 1 means that a valid solution exists
10 55 18.3 
11 66 22.0 1
12 78 26.0 1
13 91 30.3 
14 105 35.0 1
15 120 40.0 
16 136 45.3 
17 153 51.0 
18 171 57.0 
19 190 63.3 
20 210 70.0 
21 231 77.0 1
22 253 84.3 
23 276 92.0 1
24 300 100.0 1
25 325 108.3 
26 351 117.0 1
^C
It took less than a second to solve up until N=26. For N=27 – the calculation took way too long, so I stopped it. However, I spotted a pattern. It can be solved for N = 9, 11, 12 & 14 and then again for 21, 23, 24 & 26. It appears that this pattern might repeat itself for every 12th value of N. With this in mind I tested it for N=33 onwards (which is N=21 +12)
triad.exe 33
N dots triads solve
   
33 561 187.0 1
34 595 198.3 
35 630 210.0 1
36 666 222.0 1
37 703 234.3 
38 741 247.0 1
^C
Bingo! What about from N=45, which is +12 from N=33?
triad.exe 45
N dots triads solve
   
45 1035 345.0 1
46 1081 360.3 
47 1128 376.0 1
48 1176 392.0 1
49 1225 408.3 
50 1275 425.0 1
^C
Same story. Why this pattern and what about the values inbetween these? I was not 100% sure at this stage. But, with the clock ticking and some confidence I submitted my answer which got my name on the Correct Submissions list.
284, which is 2 + 9 + 11 + 12 + 14 + 21 + 23 + 24 + 26 + 33 + 35 + 36 + 38
So why does the pattern repeat and what about the other values?
We already know that N=9 is valid  as it was solved by hand. By simply adding rows of triads, we can extend the valid solutions to N=11 and also for N=12 and N=14.
Similarly, the N=21 triangle can be build up from other triangles that are all valid solutions. Starting with N=9 and adding triangles where N=9 and N=12. As per initial obervations the corner triads can be ignored if need be. This same pattern repeats every 12th row.
So the backtracking program was probably not required, but it was definitely a good way to validate the solution.
I mostly use Python for this kind of puzzles. However, similar to my solution to the February 2020 Jane Street Puzzle, I decided to rather use OCaml. This was mainly to improve my OCaml skills but also for performance.
An Array
is used to model the triangular grid which is created with the create_grid
function in the code.
Grid Coordinates
 1 2 3 4
+> xvalue

1 ! 0 1 1 1 Every 0 is a dot in the triangular grid
 Sample here is where N=4
2  0 0 1 1
 If grid value is:
3  0 0 0 1 0 > empty, available to place a triad
 1 > occupied / unavailable
4  0 0 0 0

yvalue
Triads can be placed in the grid pointing Up
(like a pyramid) or Down
(inverted pyramid). This is done by the change_grid
function. The X
in diagram below refer to the dot being evaluated. The coordinate of X
is determined by the next_dot
function. A triad placement is_valid
if all three points of the triad are empty (0).
Up Down
X X 0
0 0 0
See below for the the full program. I used the OCaml for Windows [3] compiler. If you don’t have an OCaml compiler, you can run the code by using one of the online notebooks such as https://sketch.sh/new/ml.
(* triangle is marked with 0's, borders with 1's *)
let create_grid size =
let grid = Array.make_matrix (size + 2) (size + 2) 1 in
for y = 1 to size do
for x = 1 to size do
if x <= y then grid.(x).(y) < 0
done
done;
grid
(* find next open space for triad placement *)
let next_dot grid =
let x' = ref 0 in
let y' = ref 0 in
let size = Array.length grid  2 in
try
for x = 1 to size do
for y = 1 to size do
if grid.(x).(y) = 0 then
begin
y' := y;
x' := x;
raise Exit
end
done
done;
None
with Exit > Some (!x', !y')
type triad = Up  Down
let is_valid x y placement grid =
assert (grid.(x).(y) = 0);
match placement with
 Up > grid.(x).(y+1) = 0 && grid.(x+1).(y+1) = 0
 Down > grid.(x+1).(y) = 0 && grid.(x+1).(y+1) = 0
(* place or remove the triad *)
let change_grid x y placement grid value =
match placement with
 Up > grid.(x).(y) < value;
grid.(x).(y+1) < value;
grid.(x+1).(y+1) < value
 Down > grid.(x).(y) < value;
grid.(x+1).(y) < value;
grid.(x+1).(y+1) < value
let rec solve grid =
let dot = next_dot grid in
match dot with
 None > true
 Some (x, y) >
let check placement =
if is_valid x y placement grid then
begin
change_grid x y placement grid 1;
if solve grid then true else
begin
change_grid x y placement grid 0;
false
end
end
else false in
List.exists (fun z > check z) [Up; Down]
let print_header () =
Format.printf " N dots triads solve\n%!";
Format.printf "    \n%!"
let () =
print_header ();
for n = 1 to 26 do
let dots = (n + 1) * n / 2 in
let triads = float_of_int dots /. 3. in
let result =
if (dots mod 3 = 0) && (create_grid n > solve)
then "1" else "" in
Format.printf "%3i %9i %9.1f %6s \n%!" n dots triads result
done
In my case I set the start value with a command line argument and used the Unix.gettimeofday()
function to measure execution time. This was excluded from the code above as the Sys
and Unix
module will only work on local installations of OCaml (including Windows).
let () =
let i = try int_of_string Sys.argv.(1) > max 1 with _ > 1 in
let t = Unix.gettimeofday () in
print_header ();
for n = i to 99 do
let dots = (n + 1) * n / 2 in
let triads = float_of_int dots /. 3. in
let result =
if (dots mod 3 = 0) && (create_grid n > solve)
then "1" else "" in
Format.printf "%3i %9i %9.1f %6s %10.1f \n%!"
n dots triads result (Unix.gettimeofday () . t)
done
Commands used to test, compile or profile the code.
# testing
utop triad.ml
# final version
ocamlfind ocamlopt o triad.exe linkpkg package unix triad.ml
ocamlfind ocamlopt o triad.exe linkpkg package unix noassert rectypes inline 1000 triad.ml
# Profiling
ocamlfind ocamloptp o triad.exe linkpkg package unix noassert rectypes inline 1000 triad.ml
./triad.exe
ocamlprof triad.ml
Although the program above assisted me to find the correct answer, it was very inefficient. Here are some of the obvious changes that can be made:
A comment on style. I used ‘Exceptions’ in my code  see the use of raise
in the next_dot
function. This approach is seen as not good style. As per Jane Street Style Guide:
Although raising is fine, one should not write code that depends on which exception is raised.
Another possible improvement area is not to use an Array
to keep track of the solution space.
This was a lot of fun. Thanks to Jane Street Capital for posting the puzzle. If you can suggest changes to the code or have also solved this puzzle, let me know.
[1] Jane Street  Puzzle Archive.
https://www.janestreet.com/puzzles/archive/. Retrieved: 20200403
[2] Source code for this post on Github.
https://github.com/whoek/janestreetpuzzles/tree/master/202004. Retrieved: 20200501
[3] OCaml for Windows  Graphical Installer
https://fdopen.github.io/opamrepositorymingw/installation/. Retrieved 20200501
[4] Backtracking Iterators, JeanChristophe Filliatre
https://www.lri.fr/~filliatr/publis/enum2.pdf. Retrieved: 20200410
[5] Functional Programming  Lecture 6: Folding and Backtracking, by Łukasz Stafiniak
http://www.ii.uni.wroc.pl/~lukstafi/pmwiki/uploads/Functional/functionallecture06.pdf. Retrieved 20200410
[6] OCaml  Sudoku par backtrack
https://openclassrooms.com/forum/sujet/ocamlsudokuparbacktrack
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