When you think of mechanics, science, and mathematics, do you think of excitement, beauty, and nature? More likely, you think back to some boring, vapid math class you were forced to sit through during your days in school. This couldn’t be further from reality! Mathematics, and its relation to the construction of mechanical puzzles, has a magical quality of transcending its own genre - beyond the numbers themselves! Numbers are patterns, and in the real world, they come to represent shapes and aesthetically pleasing images. Moving even further with this idea, *sequences* of numbers come to produce beautiful results, as we will show here by highlighting some of the favorites from the Kubiya sequential puzzle collections. You will see how sequences and patterns of numbers can be perfectly paired with the artistic vision of a designer to produce wonderfully entertaining and genuinely *fun* creations. Check it out, and see what you think!

**What is a sequential puzzle, exactly?**

Mechanical puzzles classified as **sequential** are that certain niche of puzzles which can be solved by advancing from a preliminary state to some objective, end-state using sequences of moves implemented in the proper order. With a sequential puzzle, your list of available moves is finite, and usually obvious. The order of moves, however, is not. The puzzler will typically have a clear idea of the possibilities, and the real puzzle is choosing which move to make next. A trick-style puzzle will not make it obvious what your options are, because (of course), the designers are trying to trick you! True sequential puzzles will typically have the available choices at any moment constrained and they would depend on the current state of the puzzle. Basically, this just means that the history of moves made up to that point will determine your range of motion at that same point. This differentiates sequential puzzles from an assembly or packing puzzle, where any piece can be used at any time. If this doesn’t make sense yet - don’t worry about it! Next up, we’re going to look at a few examples of my favorite **Sequential Puzzles from Kubiya**, and highlight what makes them a stellar choice for your own collection.

**Towers of Hanoi**

This creation is a very popular puzzle, and one you have most likely seen before! The challenge for the **Towers of Hanoi** is to transfer different sized rings from one side of the wooden structure to the other, using a middle tower as an intermediary placeholder. The puzzle has two basic rules: the first, only one disc can be moved at a time, and the second – at no time can a larger disc be placed on a smaller one. Other than that, place away, as long as the bottom disc is always bigger. Sounds simple, no? However, even the quickest solutions take hundreds of moves! The Towers of Hanoi are a great example of a sequential puzzle because once you start solving the puzzle, a pattern will start to emerge. If you pay attention, and if you choose your steps wisely, you will notice yourself making similar moves in order to continuously get closer to your end goal of moving all the rings to the other side. After a certain amount of time, all it takes is patience and endurance. The reward is oh-so-satisfying, and you can finally say that you have solved one of the most beloved (and mathematically sound) puzzles out there! In stricter, more objective terms, it meets the definition we developed earlier because there are only a finite number of moves you could make at a given time; which of the three rings will you move to which of the three towers? It’s a classic game that has been keeping brains busy for centuries - The Towers of Hanoi are a marvel of mathematics! And – you might have guessed – the solution itself can be broken down into a formula. The number of moves required to solve a tower of *n* discs is (2* ^{n}*-1), so for

*n*= 1, 2, 3 ...10 discs, you will need to perform 1, 3, 7,15, 31, 63, 127, 255, 511, or 1023 moves, without errors. Don’t worry, though, the 9-Ring Kubiya version isn’t so scary – just fun! You certainly don’t need to know anything about mathematics to play, but it’s nice to peek behind the curtains of a great puzzle once in a while.

**250-Move Schloss Puzzle Lock**

Right out of the Constantin Puzzle Collection, and available from Kubiya, up next I wanted to highlight the **250 Move Schloss Puzzle Lock**. This is a prime example of a sequential puzzle, and it’s a lock that must be undone without a key! There are four sliding, metal pins on the front that cannot move freely. So, you must move the pins in *sequential* order. As the name of the lock implies, there is a minimum of 250 moves that are required to open the lock. Just think of how many moves it takes without knowing the answer! Here you’ll find enough of a struggle to keep the most valiant puzzlers busy for hours and hours. Designed by puzzle master, Jean Claude Constantin and Co-branded with Kubiya Japan, this designer lives in Germany, where he designs and manufactures many different wood and metal puzzles, as well as magic tricks and other intellectual toys and games. Kubiya has a diverse collection of many puzzles from this prolific artist; we love displaying his hard work for loyal collectors, as well as newcomers to the puzzling world. The Puzzle Lock and the Towers of Hanoi are wildly different at first glance, with their own unique styles. However, when these two puzzles are boiled down to their most basic properties and patterns, you can see the beautiful sequences that are the building blocks for two beautiful works of art.

**A Few Final Words**

In the famous play *Arcadia* by Tom Stoppard, we hear a simple conversation between student and tutor about the nature of mathematics. The main character, Thomasina, is ranting to her tutor Septimus. She says, “Each week I plot your equations dot for dot, X’s against Y’s in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? **Do we believe nature is written in numbers?**” Well, it turns out that the world around us is brimming with sequences of numbers, whether it’s a fractal on a leaf or a sequence of levers that you pull to open a box. I love mechanical puzzles because they combine rich, artistic beauty with mathematical integrity, without the need to sacrifice or cut corners on either side of the fence. Engineering and art are twisted and blurred together until you might never know whether you’re learning, having fun, or hopefully both.

Thanks as always for reading, cheers to the start of Fall, and happy puzzling!