Monthly Archives: December 2018

LOS ANGELES: Scientists created the world’s smallest noughts and crosses board using dynamic DNA origami tiles – microscopic organic structures that can be programmed to transform into predefined patterns.
The team at the California Institute of Technology in the United States had previously used DNA origami to create tiles that could be designed to self-assemble into larger nanostructures bearing predefined patterns.
They had chosen to make the world’s smallest version of the iconic painting by Italian polymath Leonardo da Vinci, Mona Lisa.
However, the technique had a limitation similar to that of da Vinci’s oil paintings: once the image was created, it could not be easily changed.
The team has now created more dynamic tiles, allowing researchers to reshape already-built DNA structures.
Using this technique, they created a microscopic tic-tac-toe game in which players place their X’s and O’s by adding special DNA tiles to the board.
“We have developed a mechanism to program the dynamic interactions between complex DNA nanostructures,” said Lulu Qian, assistant professor at Caltech.
“Using this mechanism, we have created the world’s smallest game board for playing noughts and crosses, where every movement involves molecular self-reconfiguration to swap hundreds of strands of DNA at a time,” Qian said.
Each strand of DNA is made up of a backbone and four types of molecules called bases. These bases – adenine, guanine, cytosine and thymine, abbreviated as A, T, C and G – can be organized in any order, the order representing information that can be used by cells, or in this case by nanomachines.
The second property of DNA that makes it useful for building nanostructures is that bases A, T, C and G have a natural tendency to pair with their counterparts.
However, a sequence can also pair with a partially corresponding sequence.
The other technology, self-assembling tiles, is designed to behave like the pieces of a puzzle. Each tile has its own place in the assembled image, and it only fits there.
In creating their new technology, the team endowed the self-assembling tiles with movement capabilities. The result is tiles that can find their designated place in a structure, and then kick out the tile that already occupies that position.
To start the tic-tac-toe game, Qian’s team mixed a solution of blank tiles in a test tube. Once the board is assembled, the players take turns adding X tiles or O tiles to the solution.
Due to the programmable nature of the DNA they are made of, the tiles were designed to slide into specific places on the board, replacing the blank tiles that were there.
An X tile could be designed to only slide in the lower left corner of the board, for example. Players could put an X or an O in any empty spot using tiles designed to go where they wanted.
After six days of engrossing play, Player X emerged victorious, researchers say.
The goal of the research is to use the technology to develop nanomachines that can be modified or repaired once they have already been built.
“When you have a flat tire, you will probably replace it instead of buying a new car. Such manual repair is not possible for nanoscale machines,” said Grigory Tikhomirov, senior postdoctoral researcher at Caltech. .
“But with this tile-moving process that we have discovered, it becomes possible to replace and upgrade several parts of nanoscale machines designed to make them more efficient and sophisticated,” Tikhomirov said.

Welcome to The Riddler. Each week, I propose problems related to the things that are dear to us here: math, logic, and probability. There are two types: Riddler Express for those of you who want something small and Riddler Classic for those of you in the slow motion of the puzzle. Submit a correct answer for either one, and you might get a shout out in next week’s column. If you need a clue or have a favorite puzzle to collect dust in your attic, find me on twitter.

Riddler Express

Chess fans like to claim that there are more chess sets possible than there are atoms in the universe. Well la di da. What about us tic-tac-toe enthusiasts? How many possible noughts and crosses are there for us?

Submit your response

Classic Riddler

From Tom Hanrahan, a variation of the classic:

You are about to play a game of noughts and crosses, with an advantage and a big disadvantage. The advantage is that you can start first. The downside is that you are blindfolded and therefore are not informed of your opponent’s movements.

It works as follows. The squares of the board are numbered from 1 to 9, as follows:

begin {matrix} nonumber 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 end {matrix}

You start by calling a number corresponding to the box in which you want your “X” to be placed. Your opponent then chooses a square for his “O”, but you don’t know which one. You continue like this, alternating turns. Each time you choose a square already occupied by your opponent, you are informed that the square is occupied, and you must then choose another one until you find an empty square.

Can you design a strategy where (just like in regular tic-tac-toe!) are you guaranteed to never lose?

Submit your response

Solution to previous Riddler Express

Congratulations to 👏 Harry Posner 👏 from Boston, winner of last week’s Riddler Express!

Last week we met Louie, who lived in a town where there was a 50% chance of rain every morning and a 40% independent chance every night. Louie went to and from work each day, bringing an umbrella with him if it was raining and not bringing one if it wasn’t raining. Sunday evening, two of his three umbrellas were with him at home and one was at his office. What were his chances of getting through the work week without getting wet?

The odds of Louie staying well dry were exactly 69.27%.

The author of this riddle, Josh Vandenham, suggested three ways to get the answer. The first is to proceed by brute force through all possible weather conditions, keeping track of Louie’s umbrellas in each case. There are two relevant times of day, for five days, and two things can happen at each of these points (rain or shine), so there are (2 ^ {10} = 1,024 ) weather patterns possible. It is a lot, of course, but not totally impossible.

The second is to narrow down the possibilities and keep track of the different “states” of Louie’s umbrellas. He could have three at home and zero at work, two at home and one at work, one at home and two at work, or zero at home and three at work. Only four states, which is not too complicated. The probability of each state can be seen in this overview of the Nate Langhinrichs solver spreadsheet.

By the time Louie arrives home on Friday night, there is a 30.73% chance that he has been caught in the rain without an umbrella, or 69.27% ​​that he has stayed dry.

The third way to solve is to generate computer-simulated weather and travel. Solver Lowell Vaughn, for example, took this approach and was kind enough to share your Python code.

Finally, another way to approach this problem would be with something called Markov chains – random event models, which use sophisticated linear algebra tricks like matrices. In this approach, the situation (the umbrellas) depends on the previous situation. This approach has been put to good use by the Aaron Cote solver. The movement through the random states (the work week) is described by a transition matrix, which contains the probabilities that the umbrellas move from one location to another on a given day. In our particular case, the matrix looks like this:

begin {pmatrix} nonumber 0.3 & 0.2 & 0.0 & 0.0 & 0.5
0.3 & 0.5 & 0.2 & 0.0 & 0.0
0.0 & 0.3 & 0.5 & 0.2 & 0.0
0.0 & 0.0 & 0.3 & 0.5 & 0.2
0.0 & 0.0 & 0.0 & 0.0 & 1.0 end {pmatrix}

In this solution, there are five states: zero, one, two or three umbrellas at home plus the state Louie has ever been wet in. The rows of the matrix are Louie’s current state and the columns are his destination. For example, he starts from the third row (two umbrellas at home). There is no chance that he will end up with zero umbrellas at home, there is 0.3 chance that he will end the day with only one at home, 0.5 chance with two at home, 0 , 2 luck with three at home, and no chance he will get wet this first day. To obtain the probabilities for the whole working week, it suffices to multiply this matrix by itself five times, representing the meteorological hazard occurring over the five days, which gives us this:

begin {pmatrix} nonumber 0.04791 & 0.07634 & 0.04976 & 0.01776 & 0.80823
0.11451 & 0.19889 & 0.15274 & 0.06752 & 0.46634
0.11196 & 0.2911 & 0.22553 & 0.12610 & 0.3073
0.05994 & 0.15192 & 0.18915 & 0.12425 & 0.47474
0 & 0 & 0 & 0 & 1 end {pmatrix}

Finally, let’s look at the third row (the state it started in) and the last column: It’s the chance that it gets wet during the week. Subtract from 1 to find the chance that it does not have get wet and you get 69.27%.

Overall, given the extremely rushed climate that Louie lives in, his Random Umbrella system doesn’t work too badly. TGIF.

Solution to previous Riddler Classic

Congratulations to 👏 Steve Altschuld 👏 of New Albany, Ohio, winner of the Riddler Classic last week!

Last week, seven anti-social settlers moved to a circular meadow with a radius of 1 mile, where they would temporarily overcome their anxieties to work together on building houses far from each other. Specifically, they wanted to maximize the average distance between each colonist and his nearest neighbor. With seven houses to build, it was pretty straightforward: one would build their house in the center of the circle, while the other six would form a regular hexagon along its circumference. Each settler would be exactly 1 mile from their nearest neighbor, so the average distance is 1 mile. But at the last minute, a particularly antisocial settler canceled his move to the prairie altogether, leaving just six settlers with homes to build. With fewer people, could they find a better arrangement? Could they increase the average distance to more than 1 mile?

Yes indeed they could. Specifically, they could increase the average distance from the nearest neighbor to about a huge 1.0023 miles! Ah, the great outdoors.

Here’s what the Optimal Antisocial Grasslands neighborhood looks like, courtesy of the authors of this puzzle, Randi Goldman and Zach Wissner-Gross.

Note that the “center” house is only a a little bit off-center. This is the key.

The particular case of six settlers is proving particularly tricky, as Randi and Zach explained: If there are two settlers, they will simply build houses at diametrically opposed points on the circumference. If there are three, four or five settlers, they will organize their houses in a regular polygon. If there are seven settlers, they will form a regular hexagon plus a settler in the center of the circle. But six is ​​different.

Our winner this week, Steve, explained how he started his resolution: With six settlers, you can place five houses on the circle and one in the middle. (If all six are on the circle, the best you can do is 1 mile away.) I placed the outer five houses where they would go if I created a hex without the bottom vertex. I then realized that I could move the central house north and south a bit and drag two pairs of houses – the two opposite pairs east-west around the circumference, one pair above the other pair – around the edge of the circle a little, in tandem. So now I have a formula with three variables: the distance between the central house and the upper house, the angle of the upper pair from the upper vertex, and the angle of the lower pair from the upper vertex. That would be a pretty awful equation, so I did what a lot of us do: I did it on a spreadsheet and toggle it until I got my answer.

Solver Laurent Lessard presented its “modeling and optimization” approach, in which he considers various methods to solve such optimization problems, including climbing, interior points, simulated annealing and the sound very cool particle swarms. He illustrated the optimal arrangements for this problem not only for six settlers but also for all numbers from two to 21 – from simple east-west separation to complex Ritz cracker.

This excellent solution has generated an interesting Twitter chat, in which mathematical collaborations have been proposed, additional optimization techniques have been suggested, and it has been underline that an entire area of ​​academic research was devoted to such questions with regard to electron crystals. Not only that, I was alerted to what has become my new favorite website: packomania.com. This is what you think it is.

Want more puzzles?

Well, are you unlucky? There is a whole book full of the best puzzles from this column and puzzles never seen before. It’s called “The Riddler”, and it’s in store now! Consider your vacation shopping done.

Would you like to submit a puzzle?

Email me at [email protected]