I am always sometimes trying to find activities for all four kids to participate in together, with their varied abilities.
There is an old transport puzzle that I caught a video about a while back sometimes called the Bridge and Torch Puzzle. It is usually a mental puzzle, but given the four characters in it, I thought we could act it out.
I showed the kids the first part (up to 2:00) of the animated video below to explain the premise and rules of the activity.
Then we biked to a walking/biking bridge not too far from our house.
Then, because my kids wouldn’t take 1,2,5, and 10 minutes, respectively, to cross the bridge, I timed each one running across. To get a better sense of your runners’ speeds, you might want them to run a round trip and cut the times in half (to account for transitions). This activity is best if your 4 runners have clearly different running abilities (our 3, 5, 7, and 8 year old crew was great for this).
To calculate the goal time: add the fastest and slowest times and then add 3 times the second fastest (these are one-direction times). You might give a little bit of wiggle room time, but not so much that a different (less-than optimal) running arrangement would succeed.
Their first try was a very common wrong attempt, but then we talked through where they could make up time. The second attempt was a better arrangement and saved them more time.
Watch the 2nd half (after 2:00) of the animated video for the solution, if you want.
With Tiffany working full-time I decided to move her desk upstairs and transform that space into a reading nook. Surfing the web for reading related art projects, I found this amazing piece by Ekaterina Panikanova.
I can’t draw or paint nearly at the level I would want for this project, so I needed a printable workaround. I also did not want to have to deal with the weight of real books hanging on the wall. With those things in mind I did the following:
I went to a couple campus libraries and perused the shelves for what I considered interesting books to function as the background to the picture. Had I not rushed from idea to creation, I might have given more thought and care to the book selection, but I wanted to make it sooner than later.
I used a flatbed scanner at my university on a higher quality, color setting. You don’t want to use black & white, as much of the real book detail/texture is lost. I scanned quite a few more than actually got used. Again I might have better selected out of my scans but I was antsy to make it. (I am also a PhD student)
I suppose the image could have been of anything, but I liked the idea of the picture in the reading nook being of them reading. I got the younger girls out of their skeleton costumes, tied Aly’s hair back, and posed them with books they actually like to read. They were actually really great about the photo shoot. I took several different poses but felt like the final selection fit the shape of the space the best.
You can view the video for more explanation of the digital processing after the scans and focal picture are completed. I apologize that its quality is terrible; my computer was not interested in cooperating. (you could probably watch in 1.5x or 2x speed)
Since I’m a really high-end graphic designer/photographer (/sarcasm), I did all my editing in PowerPoint. I removed the background, adjusting as needed. Then I went with a sepia color system (for the background books as well), increasing the brightness quite a bit on the photo. Of course, you are welcome to adjust the colors to your tastes as well.
I then imported scans pages, resizing and laying them over the picture. I wanted a more gestalt feel to mine (more empty spaces) compared to Ekaterina’s (though I wanted the kids’ faces to be mostly clear). The sizing of the page scans is an important part. I needed my horizontal distance to total very close to 50 inches. I also wanted each of my scans to print on an 8.5 x 11 sheet of paper. This constrained the layout of the books. One could make bigger books but this would complicate the resize.
Once the books were appropriately sized, I brought the focal picture in front of the books and gave it a 50% transparency. I then calculated the scaling factor from the largest book page to the printable size to prepare print-ready pages. [This section is clearer while viewing the video, I think]
To make print-ready pages, I duplicated the slide and removed all but one of the book pages. I then cropped the focal image to the edges of the remaining book. I then grouped the images and resized, using the factor I had previously calculated. I then repeated that process for the remaining 11 books.
I printed in color, using a decent but not fantastic printer at the university (certainly not photo quality). I had to export as a pdf, if I remember correctly, so that the transparency would be recognized and print properly. So, before printing, make sure your prints are rendering the transparency (if you want it; I personally prefer it).
For the book’s cover, I laid each print on some roughly 1/8 inch corrugated cardboard and cut around, leaving a hardback book-sized margin.
To give the piece a more authentic feel I added additional pages behind the prints. This helped bulk out the ‘book’ and give it some curvature. I did this by laying the prints on a phonebook and cutting around them several layers deep with a utility knife. I then gave them all a soft fold together (the scans I had gave the illusion of a fold some already).
With all the pages stacked together I ran a short bead of hot glue along the edge and then flattened the glue into the pages, so they would stick together without the glue showing. I just repeated this as much as needed to get all the pieces to stick together. I then glued it to the cardboard, trying to flex the pages a bit to give it a more book-like shape.
Lastly, I laid them out, and then starting at one side, leveling, I nailed each into position, minding the alignment of the images and the needed gaps.
This project was a pretty quick one. Since we have six people in our household, I wanted to personalize a six-person game. The first to pop into my head was the classic Chinese checkers. I had originally planned to have each of us make marbles out of polymer clay colored to each one’s liking. Maybe we’ll do that another time. We went with our pictures instead this time.
First we bought some glass pebbles from the dollar store (the consistency of size and the clarity weren’t great, but the price was). Using powerpoint, I circularly cropped headshots of each of us, sized them to be a bit smaller than the cross section of the pebble, and printed them. I could have gone smaller; the pebble magnifies and the smaller version may look better. If I would have reduced the photo size I might have made a colored ring around the picture to make it stand out. I’m sure a photo-quality printer would have been nice too.
From there I just glued the pictures I had cut out onto the pebbles with white glue.
Especially because we have two girls who look very similar and, at a glance, it wasn’t so easy to recognize the images, I wrapped each with a small rubber hair tie of a favored color. I used a little smear of glue on them as well.
And that was it for the “marbles;” some tedious (60 total pebles) cutting and gluing and such but nothing challenging.
I used this template to make the board. I made the diameter of the circle match the size of the larger pebbles. I used powerpoint and simply created a circle the right size and then resized the template to match. I also cropped the image to just one point of the star.
I then taped it on my cardboard and made shallow cuts with a utility knife for all the circles. I moved the template, overlapping one row to make the next star point. And again for all six points.
I peeled the top and corrugation layers from the cut circles. I then lined out a hexagon for the top of the board. My top cardboard piece was pretty flimsy, so I glued it to a slightly larger circle, making the corrugation of the layers perpendicular.
And it was done.
We tried to play a six player game with it, but the 2 year old wasn’t interested for too long and eventually siphoned off the five year old and later then six year old. Maybe in a couple more years.
I thought about the 4x4x4 snake cube and quickly realized that the 64 cubelets could potentially also make an 8×8 square.
I searched online for anyone who had done it. Finding no one, I set my mind to finding a way or finding that it was impossible.
How could I figure out a snake arrangement that could form a square and a cube?
Calculating the number of different snakes from 64 cubelets was easy enough. There are 3 kinds of cubelets: the ends, the straights, and the turns. There are 2 ends and then each of the 62 remaining could be straight or turn, giving 2^62 (or 4,611,686,018,427,387,904, over four quintillion) different 64-cubelet snakes. Of course most of them wouldn’t make a square or a cube, let alone both. With some research I found that a 4x4x4 had 27,747,833,510,015,886 possible Hamiltonian paths. This counts rotations and reflections as unique, which was not of concern to me, so I was only looking at about one quadrillion possibilities. As for squares, I only had to look through 3,023,313,284 (really only about 378 million discounting rotations and reflections). Once I discovered these numbers, I knew a few things.
I was going to need to write a computer program (and I’m no programmer). I should start with finding paths on the square (it’s the smallest big number). If there wasn’t any overlapping solution, it would be incredible (maybe there’d be an article to write on that).
As with most work, the write-up makes things sound a lot smoother and more orderly than they really were. There was massive wasted effort exploration coming to the conclusions I did. It required a lot of learning split up over a long time (I’m getting a PhD in education policy, not snake cubes).
It ended up being possible to find a snake arrangement that produced a cube & square. However, in my pursuit to built the first one I found I left unfinished another aspect of the search I would love to figure out: how many solutions are there? I would love to have a complete map of all the snakes that solve both ways and all the different ways each snake solves. It is possible that one snake arrangement might have, for example, four cube solutions (ignoring rotation/reflection) and ten square solutions while another snake has unique solutions for both. Of course, with the carving out I’ll discuss later, another layer of matching is introduced that would make multiple solutions unlikely. But for snake cubes themselves it would be interesting to list the snakes by solutions spaces.
I would also be interested to see what percentage of hamiltonian paths for the 8×8 and 4x4x4 make it onto the list. Based on my work so far, I’d say it’s a small percent.
8×8 Hamiltonian Path Creation
So I wanted to find hamiltonian paths for the square. But with billions to work with, how was I to search all paths without missing, repeating, or wasting time?
Here’s a video of the algorithm that I finally landed on for finding hamiltonian paths on the 8×8: (I’m kind of a slow talker, so you might want to watch at 1.5 or 2x)
Basically it adds a square at a time by
Finding the neighbors of the end of the path
removing any options that obviously don’t work
and stepping back when no options exist.
It is a depth-first search, trying to find the longest path before moving to the next.
The algorithm made the snakes alongside the square paths. Sometimes different paths used the same snakes.
After finding a bunch of paths on the 8×8, I needed to see if any of the snakes formed by those paths could also make 4x4x4 cubes. I suppose I could have made the program run each 8×8 as it came along but I didn’t, perhaps the batching saved some computation because I removed repeated snakes (those with different paths but same snake configuration).
Basically it functions much like the 8×8 builder but selects the options based on the cubelet type. It starts with the first snake and the first position. It tries all the options, tracking any full paths produced. Then it proceeds to the next position and so on. Then proceeds to the next snake.
Finding the One
After making a whole bunch of 8×8 paths, I grabbed a few and eventually found a snake that fit the criteria. I double checked it to make sure it worked. Below is one solution in each form (I’m not sure if there are others for this particular snake)
It is very common for snake cubes to utilize a checkered pattern. Here I had a snake cube that could form a chess board in its 8×8 form. I thought it would be especially awesome if in its cube form all the chess pieces would fit inside. For this to be worth it to me, I needed the pieces to be appropriately sized relative the board squares and I needed all the faces showing on the cubelets in both forms to be smooth.
Eventually I came up with an arrangement that seems to work. I learned 123D Design so I could construct a virtual prototype and I’m now working on a physical one. After a rough draft I may have to produce a high-quality one.
This relatively simple idea became a consuming project that led me into learning from several fields.
This is pretty straight forward. Here is one basic tutorial. My kids love making all kinds of characters, but I thought having their characters in the image would be more interest keeping.
3. Create ‘Find the Differences’ puzzles
Create a two panel comic.
Design the first panel using a lot of adjustable objects.
Use the duplicate panel button
Make alterations in the second panel. (You may or may not want to keep track of the changes)
If the want to reuse a comic, you can remix after you save.
It’s that easy.
You know I love a surprise party, especially if it’s themed. This year, playing off 32, which is how old Tiffany is turning, I went with a 2^5 theme. In this context that means packing in 5 “party of two” dates in one day.
Thank you to all the people who took care of the kids so we could get out so long.
Lunch at Nitty Gritty (the Official Birthday Place)
and a game at I’m Board!
Swing by Sonic on the way to Fired Up Pottery
Date three: Strolling to Picnic Point and out onto Lake Mendota for the sunset
The planets aligned a couple weeks back. My wife was out of town for a couple days, which meant lonely evenings and less concern for tidiness: a great combination for a large craft project. This coincided with move-in/out weekend at my apartment complex, which equated to substantial amounts of cardboard in the recycling bin, which I gladly rescued.
We have been needing a solution for the toys that find themselves downstairs. Upstairs we still use the Tetris shelves I made a couple houses ago. They are still holding up.
I’ve been on a bit of a cardboard kick lately [see some previous posts].
I figured the cubbie-hole style would allow the cardboard to hold more weight than trying long shelves. The holes needed to hold toys so I made them 14-inch cubes (theoretically; more on that below). I estimated that 4 corrugations thick would be about 5/8 inches and would be sufficiently strong. You can see from the sketch the form of the six internal interlocking pieces and the four external pieces.
I wanted to alternate the locking because I thought it would be sturdier, but I didn’t anticipate how difficult (but clearly not impossible) piecing together would be.
Most of my pieces ended up a bit fatter than 5/8 because I found most the large boxes used doubled corrugation. Therefore I ran the doubles on the outside with matching direction and used a single in the middle with opposite corrugation direction. I think the design would hold the pieces together with no glue, but gluing makes cutting much easier. I used hot glue, but I wouldn’t recommend it. Some white or wood glue would be better, I think, but I didn’t have it on hand (plus, it dries more slowly). To keep the shelves square I needed to fix in two bracing pieces 14×14 inches with 5/8×2 inch tabs, as shown in sketch. I ended up just finding some very stiff cardboard and going with a single layer (same for the front pieces (though the front is not so stiff)). The back pieces I glued in. I added some glue to the external boards as well. I think longer connectors would have helped and could be trimmer later if need be.
The personalized covers were fun but can be a bit tricky to put on. I wanted covers so the kids could have some autonomy in storing their junk (they collect a lot of junk) without Tiffany and I needing to look at it. Because my boards were thicker than 5/8, I cut the front squares 13 7/8 so they would fit better and notched the boards ¼. The big kids chose their fonts from a list of stencil fonts and told me the color they desired (I had to mix the Princess Else-inspired light blue).
Whereas some of my projects span several days, weeks, or even years (from concept to completion), the 3 designs featured here hardly take any time at all if you have cardboard, packing tape, and something sharp.
Depending on the care (read: time) one wants to take and the aesthetic desired, cardboard projects can greatly range in look. Don’t write these off just because of the quick and dirty examples here.
1. Paper Organizer
This project went through two iterations. In the first the shelves were only taped in. After overloading it with projects and the occasional child using it as a ladder to get to the counter, it needed some repairs. The second version has slots cut in so the shelves poke through and are more stable.
2. Toy Food Cabinet
The kids had a bunch of small items that needed stored such as food and dishes to go with their toy kitchen, so I put together this cabinet and drawers. It has held up about a year so far.
It has closable cabinet doors on top that open to two shelves. The hold by tabs cut into the doors that match to slits in the middle shelf. The bottom housing two removable drawers.
3. Bed-side Table (x2)
The first one was made for Tiffany to put her glasses and bedside books and whatnot in. We had less than a foot of space beside the bed so I put this together to fulfill the need in the space we had. It has two removable drawers in front that fill half the depth. The top of that section is reinforced so a drink can be placed on it. The back half opens from the top and provides a nice space for larger items.
Later my oldest daughter wanted one of her own. She didn’t have the space restrictions, so I opted for a slightly larger version made from a diaper box. That way I had less cutting. I also tried to size the drawers more carefully.
As you may know from the previous Modular Origami Bedlam Cube post, an anniversary a couple years ago was Fibonacci themed. While exploring gift ideas to make, I thought of or stumbled across (I don’t recall, but I’m certainly not the first to think of it) the play on Fibonacci sequence, Fibonacci sequins. I decided to put this to work on a cube I had already made for another purpose.
The cube is an old t-shirt cut into two 1 unit by 3 unit rectangles and held in the shape of a cube with fabric glue baseball-style and dryer lint stuffing.I made them mostly inside out like one would if one was sewing a pillowcase or something.
The die features “1,2,3,5,8,13” in glued on sequins. I opted to start with 1 and 2 because it allowed me to get to the 5, 8, 13 portion of the sequence, which was the important part for the anniversary.
Because half the numbers are higher than the traditional 1-6 die, you might try it for games you want to speed up. It is also soft, so it can be used when harder dice might be too loud.
Finally, after at least a couple years after the concept popped into my brain, I have completed the Magic Square/15-Puzzle Bookshelf Covers. Before we moved to Madison we picked up this Ikea bookshelf. Long before we got one, the 4×4 design immediately led me to think about the two aforementioned number puzzles that utilize such a grid.
I collected a whole bunch of cardboard scrounging through our apartment’s recycling bin. Each square is two layers with the corrugation crossed to increase the strength and magnets glued to the back.
I wanted a solution to attaching them to the bookshelf without damaging it, so they could be removed later if we wanted.
I finally decided to go with can lids with holes poked through like buttons, stirring straws affixed to the back to better hold them in place, and then tied around the back with fishing line.
Back when Asante was starting to get into chess, I decided to try out this riff on my theme of paper pulp and puzzles. Similarly to the Bedlam Cube (because they both have 64 units), the 8×8 pentomino chess board uses 12 pentominoes and one tetromino, a square in this case. The difference is that because these pieces are effectively 2D, there exist only 12 pentominoes total. Also, because they need no specific thickness, I made these pieces only half as thick as the pulp units featured in the soma cube and snake cube. This allowed them to dry much faster.
I checkered it with a small amount of watercolor paint. It is made out of paper, so getting it even moderately wet is a bad idea (unless you are recycling it into something else).
This is a challenging puzzle without the checkering, but with checkering it can be brutal. If you are just looking for a quick game of chess, you may want to memorize or write down a solution.
To go along with the board I made some abstract chess pieces from a mix of paper pulp and homemade playdough. The playdough serves two purposes: it shapes better and gives a bit of weight. Also, it seems I set these up on the other side of the looking glass, because the queen and king have swapped places.