Mechanical Puzzle Designas a

Man-Machine Collaboration

Pavel Curtis

Pavel’s Puzzleshttp://www.pavelspuzzles.com

Overview

• Explore development of my puzzle design process

• How I did (or would) design three specific puzzles

• Lessons I’ve learned along the way

On the Problem of Spoilers

• How to describe my process without spoiling puzzles?

• Approach #1: Some designs are hard to spoil!

• #2: Spoil somebody else’s puzzles!

• #3: Spoil a puzzle that’s currently unavailable!• And hope you’ll have forgotten by the time it is available…

Birth of a Puzzle Design

• Summer 2000: International Puzzle Party

• To be held in Tokyo, Japan

• The Japanese flag is very simple, elegant: dissect it?

• Break it up into 12 ‘domino’ shapes

• A 4×6 tray yields 12 pieces: that’s a good number

• Feels way too easy…

• Make the pieces double-sided?

• It should have a unique solution, right?

Hinomaru:The Japanese Flag Puzzle

Good Idea! Now What?

• Is it possible to design pieces with a unique solution?

• Time for software!

• Compute all possible ways to dissect a 4×6 rectangle into twelve 1×2 domino shapes• Answer: something like 261 distinct dissections

• Manually select the one that looks most ‘random’

• Assign faces to the backs such that solution is unique• Lots and lots of assignments work: how to choose?

• Try to use each face design the same number of times?

Design Complete!On to Test Solving…

• I abuse my friends.• A lot.

• The prototype:

• First tester (a genius): 2 hours and 30 minutes!

• Second tester (super-genius): 45 minutes!

• Making it hard enough was not the problem…• Can it be solved without exhaustive, brute-force search?

Lessons Learned I:Pavel’s Three Laws

• It’s easy to design a hard puzzle.

• It’s hard to design a good puzzle.

• It’s good to design an easy puzzle.

(At least sometimes)

Lessons Learned II:One Useful Design Process

• Imagine an interesting puzzle ‘story’

• Use software to search for puzzles that tell that story

• Almost always, only two possible outcomes

• No such puzzles? ‘Open up’ the story

• Too many puzzles? Add more detail to the story

Example #2:‘Four Sleazy Pieces’

• Designed by Stewart Coffin• American ‘elder statesman’ of puzzle design

• Let’s pretend that I designed this puzzle• A non-spoiling analogue for my ‘Sleazier’ puzzle

• It was inspired by my (failing at) solving Stewart’s puzzle!

First, the Story

• Pack a square tray with polyominoes• Polyominoes = shapes made of multiple squares

Make the Tray a Weird Size

• Not an integral side-length:

• Experienced puzzlers knowwhat this means• Anything weird is that way

for a reason

~5.8

units

Then Double-Cross Them!

• Pieces do go in at an angle, but not that one!

• (It is called ‘Four Sleazy Pieces’…)

Time for Software!

• Find all ways to dissect the goal shape• Reusable module #1: shape partitioning algorithm

• For each possible dissection, perform solving tests• Reusable module #2: shape solving algorithm

• Must solve goal shape in exactly one way

• Must solve these shapes in no ways:

Drowning in an Ocean of Puzzles!

• Thousands or even millions of dissections work!

• Add more constraints to the story:• No straight pieces

• No duplicate pieces

• All pieces about the same size

• Still too many? Talk to my wife!• a.k.a, my ‘sadism consultant’

• “Can you make it almost solve 5×5?”• Pieces must solve these shapes:

Puzzles that are Stories

• A ‘story puzzle’ is solved in ‘chapters’

• Goal of first chapter is (fairly) obvious

• Solving that chapter leads to a new puzzle• Welcome to Chapter Two!

• Final chapter yields a satisfying conclusion• Typically an appropriate word or phrase

• Story puzzles usually have 3 to 5 chapters

Example #3:Holiday Discount Puzzle 2012

• Genesis: Can this shape make a puzzle?

• Eight square pieces,each with or without certain corners

• Need solution to be unique• Ensures a consistent ‘next chapter’

• Answer: create some kind of picture

• Pieces must have ambiguous placements

Chapter One

• Eight pieces:

• Goal: make a solid diamond:

• Each piece could fit attwo corners and two edges

Chapter Two

• Somehow read a message off the assembly

• Thus, need letters on the pieces:

• Solver must infer: read clockwise around the diamond, ignoring other letters

T H

A

E R

SE

E S

IL

T A

TE

D W

ON

E

AR

S

PR

C E

ER

Chapter Three

• Message suggests reading the other letters• But doesn’t explain exactly how

• Solver must infer: read left to right, top to bottom

• Result: “ANSWER IS _______”

• Only three chapters here• Want the discount puzzle to be on the easy side

When Software Can’t Help

• Many puzzle searches can’t be automated:• Disentanglement / topological puzzles

• Trick-opening boxes

• My “Get a Clue!” puzzle

• This category is growing for me• Still lots of design lessons to learn!

Story-First Design

• Craft a pleasing solving story

• Use whatever tools work to search forpuzzles that tell that story

• Revise or refine the story as indicated

• Treat test-solvers as story critics

What Story Would YouLike to Solve Today?

Thank you.

Pavel’s Puzzleshttp://www.pavelspuzzles.comHigh-quality mechanical puzzles, direct from the designer!