Puzzle Archive

April 2020 Solution

The answer to this month’s puzzle is 284.  All numbers congruent to 0, 2, 9, or 11 mod 12 are able to be constructed, and no others are.

We can show that all of these numbers work.  The arrangements for N = 2, 9, 11, and 12 are presented here.  We can then add 12 rows to any of these 4 triangle sizes by adding an N=12 triangle, and a 12-by-N parallelogram, below them.   We can construct those parallelograms by adding together 2×3 parallelograms until they are of the dimensions 12-by-N.

Proving that no other numbers work is… significantly more difficult (and part of the reason we cut things off at N=40).  We didn’t realize this when we came up with this puzzle, but it turns out this problem has been investigated before by none other than the great John Conway, who the world sadly lost this past month.  What a fortunate, if unintentional, tribute to Professor Conway.  Here are links to his writings on the topic and to the OEIS page devoted to this sequence.

Congratulations to everyone who solved this month’s puzzle!

Correct Submissions From:
Jakkap
Miklós Horváth
Magister Mugit
Monotof
Glauber Guarinello
martinobdl
kinder3141
Ethan Morgan
David Fair
Guillermo Wildschut
Tai Zong
Primusa
juxd
Sébastien Geeraert
Nima Oparvar
Sergey Tychinin
Willem Hoek
Kendric Postrero
Sean Egan
Jordan Katz
Gordon Macshane
Rahul Saxena
tongssss
Sabster
EP
Dimas Ramos
Juan Casado Ballesteros
Keith Schneider
Daniel Ramos
zhelih
Magic Mister
Tomer Tzadok
Archie Almond
Xizhe Cheng
Jaime Benabent
George Paraschiv
Dewei Chuan
Konstantin Vladimirov
Kristof R.
Scott Okuno
Zihao Li
Kristof S
VenuVGopal
Matthew Monjarrez
David Wolfe
Mikhail Soumar
Ezio
Karl Mahlburg
Christopher Kei
Stranger
Nigel Brumpton
Edward McFarlane
Erik-Cristian Seulean
Geoff Donoghue
Prateek Pinisetti
Sandip Ghoshal
Raashi Tapadia
raxu
PB
Walter Sebastian Gisler
Guillaume C.
John Bamford
Hudson Vieira
Eugene Du
Malak Campbell
Ming Wang
Mathew Varughese
Caroline Norman&Sasha Hydrie
Alex Wice
Lyudmil Kouzmanov
farlab
Gabe Chalom
Calum Heraty
Edward Gathuru
Yoma
HonorableNed
Alok Thatikunta
marios georgiou
lll
sniffleheim
Jeffrey Fu
Senthil Rajasekaran
Ohad Rau, Zach Baylin, & Eric Frankel
Will Gulian
Sandman
Anirudh Raghavan
Edgar Wang
Alex Kalbach
Kevin Becker
Will Christerson
OSINT
Burnside and Cauchy
Michael DeLyser
Andrew T.
Aaditya Raghavan
MrPapaya
Will Liao
vgamula
sjs
Garrett Andersen
Wula
MK
Andi Qu
Adam Hodgson
Angus King
Pat Gardner
Ben Reiniger & Mark Taylor
Orlin Kuchumbov
Clayton Bass
arriopolis
Stan Tendijck