Finding Prime Numbers with Leaping Frogs

Imagine a pond that contains a long line of lily pads, each labelled with a number, starting with '2', '3', and so on.

A scary stork enters the pond, looking for frogs!  Each turn, the stork will move to a new lily pad, starting with lily pad 2.

When the stork reaches a lily pad, it looks for frogs:

• If the lily pad has no frogs on it, the stork finds one in the water. Take a new frog token and write the lily pad's number on its back. Put it on the lily pad with the stork.
• If the lily pad already has frogs on it, the stork won't look in the water, so this lily pad does not get a frog of its own.

The stork then shouts, "Boo!"  This scares the frogs, who jump away. All the frogs jump forward a number of lily pads equal to the number written on their backs.  Frog #2, for example, will make the first leap from lily pad #2 to lily pad #4, and then later to lily pad #6. (It's okay for frogs to share lily pads.)

Stop after however many turns you like.  When you're done, the numbered frogs represent all the prime numbers that you found (at least from numbers 2 to whatever lily pad you stopped at).

When a stork first reaches a lily pad, all the frogs there are the prime factors of that number.  For example, when the stork reaches lily pad #20, the stork will find frogs 2 and 5 there.  (20 = 2 x 2 x5.)  Lily pad 19, however, is empty when the stork arrives arrives: the newly minted frog 19 indicates that 19 is a prime number, and there are no other prime factors (other than 1).

To summarize:

1. Move the stork forward one lily pad (on the first turn, place it on lily pad 2).
2. If the lily pad has no frogs on it when the stork gets there, make a new frog with the lily pad's number.
3. All the frogs on the stork's lily pad jump forward a number of pads equal to their number.

Fiasco with Kids

As I hadn't managed to get my relatives together to play Fiasco yet, I figured I'd give it a whirl with my kids, and it was surprisingly fun and accessible to them, despite their young ages (5 and 8).

Creating the character relationships went very smoothly - rather than having them browse the playset lists, we just took turns rolling and took what we got. (Although I did edit out the untoward objects and the needs about getting laid!)

By the time we were done, Leah (5) was a veterinary surgeon (Dr. Daisy), locked in a rivalry with Morgan's character, Dr. Eggnog. (A certain gonzo element creeps in with kids!) My character, Burt Cummings was a vet school wash-out, having left when bowel problems made it impossible for him to afford tuition. Now, he's getting illegal surgeries done via Dr. Daisy (professional/client relationship). Our location was (of course) the Animal Care Clinic on Center Road.

We kept the scenes super short and punchy, so the whole game was done in less than an hour.

My girls play 'pretend' all the time, and what's neat about it is the constant out of character negotiation they do about their roles and the upcoming events. As a result, Fiasco's scene framing seemed very natural for them, and they rolled easily with whatever Resolves were handed to them.

After the end of Act One, we rolled two Twists, "Dangerous Animal (Perhaps Metaphorical) Gets Loose" and "Ugly Struggle Ends in Death". The former we played literally; Dr. Eggnog accidentally releases a rabid dog while hunting through the vet clinic for some dirt on Dr. Daisy; later Burt runs into it and gets hurt. I then showed up at Dr. Eggnog's place with a shotgun, looking for revenge (long story) and Eggnog ends up biting it, which Morgan was totally into.

We still had a few scenes left, so we followed the game's advice and played Eggnog's last scene as a flashback - Burt and Eggnog back in vet school, just chatting. There was no real conflict, but I found the juxtaposition between their mortal struggle and their earlier friendship surprisingly poignant.

Fiasco is a very lean game, and the mechanics are a lot simpler than the rules make them out to be, somehow - there's so much advice about how to apply them that the learning curve looks steeper than it is, partly because the rules are spread out through the fairly long text. The summaries are handy, but only as memory-joggers - "do some dice math". The whole game (playsets aside) could probably be printed on three pages, with all the advice following.

All in all, it was slightly too involved for Leah, whose interest waned when there were about three scenes left but Morgan said she'd play again without hesitation. Fun!

Centaur Goes to School

Here's a video of the centaur in motion!

Centaur Nears Completion

Here's a new set of pictures of the completion of the centaur body!

As I posted last time, I'd resolved to make a new body out of foam. The first time around I didn't have quite enough foam, so I thickened it with alternating slices of a thinner sheet of foam I used to pad the harness.

The problem was that the surface along the torso was all irregular, and carving wasn't very easy with the different foam densities. I'd made a whole new skeleton, so why not make a whole new foam torso?

As before, I stacked the foam segments on the spine, but this time I glue-gunned them into place.

Next I carved the out the legs. I used the skeleton as a template to make sure they were the right size, then started rounding them out from there using a carving knife. The carving knife was a great tool, although the foam blunts it after a couple of minutes' cutting, so I kept a sharpener handy.

The next step was to insert the skeleton. Following some advice on a puppetry blog, I made slits, then plucked out a tube-like chamber, wide enough to house the plastic leg bones.

Then, reattaching the hip bolts, we get the assembled rear legs.

A few more screws to reattach the hips to the aluminum spine.

To carve the torso, I kept a small plastic toy horse at hand (maybe you can spot him in the background). Nevertheless, I made two errors - one, not starting out with enough foam to make the round stomach, and secondly, I made the whole thing too long. I hadn't accounted for the length of the ass - the haunches protrude well beyond the foam skeleton.

Overall the net effect, particularly striking once Morgan was suited up it it, was a ridiculously long horse body, making Morgan look minuscule compared to the school bus sized body she was dragging around. At this point, though, I had no objectivity and couldn't really see it, but my wife Danielle was pretty insistent it looked silly.

A couple of inches off the spine, and it looked much better. The next thing to do was to turn it upside-down, and hot-glue on a thick slab to be carved into the stomach. This was really tricky as it was hard to plane the undersurface into something flat enough to glue onto - hot melt isn't very strong, so the final form involves two duct-tape 'belts' for strength.

Then I carved out two channels in which to mount the leg-string guides.

A quick test with Morgan. You can see in this picture that I've carved away the fronts of the thighs. I had planned to do this all along and mount them with bead-filled bags. After getting a shape I was happy with, I was very reluctant to do this step, but it was necessary as the foam was far too rigid for Morgan to be able to bend it - all the leg action was constrained to the lower joint. In the end, though, exhaustion set in and I never did fill the missing space with anything.

Next step, the skin. I haven't sewn in 23 years, not since grade 8 home economics. I was a little scared, then more gravely worried when the sewing machine started spewing out little spider-like knots of thread. But with some expert guidance (I had threaded the machine incorrectly, throwing the thread tension way off), I was on my way.

First step, some pants for Morgan. I was sewing heavy spandex, so my pattern could be dead simple; I didn't even bring the leg segments to a point to come together in the crotch, but fortunately the material is stretchy enough to accommodate despite this.

I had experimented to see how stretchy it was - about 180% stretch on one axis, maybe 130% on the other, and left what I thought would be enough room for the pelvis padding I'd carved (not shown) to make Morgan's hips look more like the chest and shoulders of a horse. But I underestimated how tight that would feel - much bunching and itching and butt crack sticking out, and so the fake muscles are on the shelf for now, seeing if I have time to make some brown, looser overshorts to hold them.

Errrrrmum-mum-mum-mum-mum-mum-mum.

Next step, remedial drapery. I wanted the skin on the horse to be tight, so following some advice from mum I just draped the fabric over it, and cut out likely-looking pieces, pinned them, then did some more.

This worked out very well, although I'd choose completely different seam placement were I to do this again. Morgan's dress will drape over the torso; the key area to make smooth is the rump. One seam up the middle, and darts wherever else necessary.

The inner thighs were the hardest part. A washable marker was my friend here (chalk pencils barely work on spandex) to label which pieces need to be sewn together. When it's off the foam body, the skin is just a floppy confusing mass of shiny fabric, it can be surprisingly hard to keep straight what needs to be sewn to what.

A marathon stretch at the sewing machine (about 6 hours) and the legs and body was finally done.

This step made me laugh - trying to 'dress' something this large was like trying to dress a sleeping person. It felt like it needed a name at this point! Some sports socks were very helpful in smoothing the passage over the very grippy foam, but the hips were a major pain.

Ta da! This is the horse with a temporary tail - we've got some blonde wigs we're going to use instead. Sorry for the visual clutter in this shot, my living/dining room was buried in bits of foam and tools, there was hardly anywhere to move.

Walking Test #1

Here's a short video of a walking test. It's very wobbly at the beginning, but the end is better. We'll see how it works once I get some rubber treads on it.

Centaur Details

This weekend I took the plunge and rebuilt the centaur skeleton. I mulled this over for quite a while, but after talking some options over with a friend, this seemed the best way to proceed.

Here's the Mark I centaur skeleton - padded aluminum harness, with everything else made of 3/4" plastic pipe. The body shape was made from layers of foam, shaved down with scissors.


If the shape looks too skinny at the back, that's because the thighs go on later.

In this version, the legs (only half of the legs are shown here) are passively moved - like a marionette, except from below. The legs would dangle from the centaur hips, while the hoof would be fixed to a ski-like slat attached to Morgan's ankle by a sort of padded manacle.

The first walking tests with the ski-legs were successful, but were unsatisfying on two counts. One, they're uncomfortable. Morgan's well aware the final result might be hot, itchy and confining, but my experience with the spider legs last year taught me that skis can be trouble in an urban environment - particularly curbs.

The second problem is, well.. horses don't walk like a marching band, with left and right legs in synch. The illusion of horsiness would be much more convincing if Morgan's right leg controlled the horse's left, and vice versa. I thought I could pull this off with one arched ski and one flat ski, but I was having second thoughts.

Spring-Loaded Legs

Taking an idea from the Walking with Dinosaur 'making of' videos, I had been mulling over the idea of using bungee to create artificial tendons. The tendons would keep the legs tense, and Morgan could pull them out of position using strings. String goes around corners nicely, so I route them near the body of the horse! The lack of skis would certainly also improve the illusion.

But the first problem with this is that the good old Mark I centaur's hips just aren't that strong. The spine and hips were connected by a single stove bolt, and I was worried the joint might give on Hallowe'en night. Deep inside the costume, there would be no possibility of repair - a sad end to Trick or Treating!

Artificial Muscles

I mentioned the missing thighs before - my plan is to make them out of spandex bags, filled with polystyrene beads. Just like real flesh, these would squish out to the sides when compressed, rather than just contracting like foam.


I sew like a five year-old at the best of times, and sewing Spandex truly sucks. It bunches, it twists, and it's flexible enough for the needle to stuff it deep inside the sewing machine whenever I reverse to close off a seam. Nevertheless, I was able to get the job done, albeit messily, and dipped into a second-hand bean bag chair for the fill.

Adding grommets to the corners, here's my finished thigh muscle prototype!



A Lifetime on the Hips

In fact, if I was going to replace the plastic hips, why not replace the whole spine? Days of work came apart distressingly quickly, and soon I was on my way. Here's the new lumbar (the child end of the horse's spine, where it attaches to the harness).


It just looks durable, doesn't it?

Old hips:


New hips:



Another improvement to the new hips is that the hip bolts themselves will now pass through two pieces of metal, instead of just one - this will keep the bolt much straighter, and allow a lot less wobble as the legs are worked.

Spring Loaded Legs

This part was slow. I was well aware that my sketch had a lot of missing details. There's process I've been through several times on this project, it goes like this.

1. Imagine cool, ultra-realistic motion effect (e.g. the walking motion of the back leg of a horse)
   
2. Come up with simple mechanism to achieve this effect (e.g. spring-loaded leg, with a string to lift it! Hurrah!)
   
3. Build it
   
4. Reality sets in. (e.g. A horse walk cycle has about eight phases to it, and that my one string can't possibly pull it off.)
   

In the end I opted for a fairly simple design - unlike my sketch, there's only one point of control for each leg. I realized I could spring-load the legs and hips so that the legs are stretched backwards, as if at the end of a stride. Then, Morgan's knee-forward movement could be used to lift the leg and pull it forwards. As she continued to move forward, the string tension would release, and plop the hoof down on the ground.

Here are the legs, attached to the pelvis. I reused the plastic pipe for the upper (thigh) and lower (foot) sections, and used a piece of flat aluminum for the horizontal shin. (By the way, horse's knees don't bend backwards - the knee is much higher up and nearly obscured by the massive, disc-like thigh, and what appears to be the knee is actually the ankle. Horses literally walk on tip-toe.)

The shin extends a little beyond the plastic pipes to provide leverage for the bungee. Here's a close-up of the inside of the right knee:


This close-up also shows the main weakness of the joints, the bolts themselves. I'm using a threaded stove bolt as the hinge, but the back-and-forth action of working the joint tends to unscrew the nut. I don't want to put a washer on and make it super-tight, because that would interfere with the smoothness of the joint. The carriage (smooth) bolts I've seen are too large. My makeshift solution so far has been to crazy glue the nut in place, but this makes it hard to maintain.

I intend to revisit this later. Something I'm thinking of doing is making my own hinges by cutting lengths of aluminum rod, and then drilling holes into them to put r-clips through on each side of the leg bones.

When it was all assembled, I had a pair of rear horse legs, spring-loaded in the fully-extended-to-the-rear position. I've also added two new aluminum pieces to the hips, to build out the pelvis. Those holes near the front are to attach muscle, creating the front of the thigh between the pelvis and the knee.

Here's Morgan, wearing the nearly completed skeleton!


If you squint, you can see the strings tied to her knees. The hooves are made from a 3" PVC connector, hacksawed in half diagonally. (Tedious, but simple.)


The Walking Test

In the three walking tests to date, it's occasionally so awesome that onlookers squeal with delight, and sometimes it's so abysmal I get worried the costume will destroy itself. The problems:

1. Humans swing their hips when they walk, but this causes the horse body to flail from side to side; Morgan has had to learn to walk like her hips were the front axle of a wagon.
2. The aluminum straps of the harness dig into Morgan's chest.
3. The harness isn't quite symmetrical (I suck at bending aluminum bars), so the horse has a slight tilt to one side.
4. On smooth surfaces (e.g. our laminate floors) the hooves slide around a lot. I think I can solve this by weighting the rear hooves and adding rubber grips to the bottom. I hope that will perform nicely on concrete and grass.
   

Centaur - Pictures Soon!

As a way of practicing for the larger dinosaur costume, I've been making a centaur costume for Morgan, my six-year old. So far I've mostly completed the skeleton, which is made of 3/4" plastic pipe, screwed together, and the core of the horse torso, which is made from a carved stack of sofa cushion foam.

Sunday I spent most of the day at the picnic table turned workbench, working on various ways to attach the costume to Morgan. I've tried to prepare her by telling her since the beginning that it might be uncomfortable, but I couldn't find a satisfactory arrangement of straps that didn't either strangle her, or affix the horse torso too loosely to her.

In the end, I've opted for thick aluminum straps which travel up her back and then bend over her shoulders in a rigid frame. This is a nice setup since there's nothing to fasten; she just wriggles in and she's good to go. (One surprise for me was that, because of kids' proportions, it took me a few tries to build something tight enough to sit properly on her shoulders, but which she could still get her head through!

Aluminum Bars!

My surprise discovery is how nice it is to work with aluminum bars. The 1/8" x 3/4" flat bars are both strong and light. Self-drilling screws go through it like butter, it's easy to saw (with a hack saw), and it's flexible enough that with the right corner, I've been able to hammer or bend it into crude curves in a pinch.

The major downside is the cost. A 4' aluminum flat rod costs over $10 at Home Depot. I'm eyeing onlinemetals.com, which is MUCH cheaper ($2 for a 4' length) and seems to ship to Canada.. for around $0.08/inch.

Three Year-Old Magic

This morning at breakfast, my three year-old, Leah, performed an incomprehensible magic trick. She's holding her hands behind her back, waving them around, then she's shouting "Ta-Da"! I have no idea what was going on in her mind, but she seems pretty pleased.

"Honey, you need to explain what you're doing so people can understand your trick."

To demonstrate, I make a fist and hold it up. "There's a penny in my hand, and I'm going to make it disappear!" I hadn't anything in my jeans pockets, so I'm just pretending - my fist is empty. Maybe I'll get a laugh out of it.

Morgan, my six-year old, perks up.

"Show me the penny! There's no penny in there."

After years of my lame daddy-magic tricks with pennies, Morgan now questions every step of the way. There's no way she's going to believe there's a penny in there unless she sees it first.

"You're right, Morgan," I say, "there's no penny in my hand. See, Leah, people need to understand what you're doing and believe it, otherwise the trick doesn't work."

I'm not going to fool anyone, but now I'm in the mood so I continue with my impromptu routine.

"Look, Leah, I'll make it disappear!" I say, holding up my still unopened fist. "I'm going to squeeze it down to nothing!" Loosening my grip, I poke my finger in one side, then in the other, ostensibly pressing the penny down to dust-speck size.

"Now that it's too tiny to see, I'm going to blow it out like dust!" I hold my fist to my mouth, blow through it, then theatrically fan open my fingers, showing my empty hand.

"Where's the penny?" I ask.

Leah thinks for a second. She wasn't born yesterday. She's turning four in a week, she's been around the block a few times now. She's seen this sort of thing before.

"It's in your ear!" she exclaims!

The Zero-Prep Scavenger Hunt for Kids

We were trying to decide what to do with the kids today when it occurred to me that it would be fun to have a scavenger hunt. The problem was that I hadn't prepared anything ahead of time, and the kids were already antsy enough that there wasn't time to do anything.

So I faked a break-in.

I kidnapped one of Morgan's toy mice, grabbed a handful of furniture from its house along with it, and stuffed them into my jacket pocket. Leaving a few telltale items sprinkled on the porch, I called Morgan down to investigate.

The star of the day's entertainment was our cheap electronic stud finder, which I claimed was a toy detector. It has green and red lights, it makes an authoritative-sounding beeping noise, and a sneaky grownup can control when it goes off by surreptitiously sliding a finger behind it while it's on.

Aiming the toy detector at various parts of our neighbourhood, we quickly determined that the stolen toys had been taken in the direction of ... the park!

We came back in to get sweaters and snacks for the excursion. Morgan knew something was up; she's nearly six, old enough to innately grasp the difference between reality and Daddy's silly tricks, even if she is still sometimes confused about where the border is.

Once she was satisfied that there was fun to be had, however, she was willing to suspend her disbelief, and eagerly added some decorative bedding to the tupperware container I had brought to store recovered toys.

Once I had established the premise, the stud finder worked amazingly well. Danielle and I strolled along the path, and I just had to point the thing into the bushes and off the kids went with gusto. Usually this was a bush we had just passed - a side effect of the "zero-prep" aspect of the scavenger hunt.

The kids got so excited by all of this that they started asking passers-by if they had seen the lost mouse. After an hour of this, we came across one couple who had already heard of our search from other park-goers! Isn't it amazing how society rallies around a crisis?

One of the nice things about the zero-prep scavenger hunt is that you can fine-tune things as you go. When the kids are climbing trees, chasing each other, snacking, or otherwise keeping themselves busy, you can just enjoy the scenery. But whenever there's a lull in the action, you can drop a toy or even just bleep the toy detector and off they scurry!

Here's a tip, though: if you drop a toy, remember where you dropped it. After dropping a charming, faux-wicker mouse suitcase in some bushy grass, I forgot to look around for landmarks. Or a decent landmark, at any rate. I could have sworn it was by this distinctive-looking tufty bit.. but then, you're surrounded by tufty bits. It's amazing how fast a lawn can turn on you.

And then, you have to explain why the toy detector suddenly isn't working. "Interference from the bridge?" was the best I could come up with.

Unable to bear the thought that I was personally responsible for the loss of this helpless, cherished toy, I suggested we link arms and walk through the field in a line, like you see on the TV when volunteers are helping the police looking for a body.

The parallel was somewhat uncomfortable, and we got quizzical looks from the joggers and bicyclists whizzing past us as we inched along. But never fear, the suitcase turned up and my "decent parent" status was saved.

Somehow, Morgan never caught on to what I was doing. In the end, she concluded that I must have set up all these toys two days earlier, "On your way home from work!"

My only disappointment was that the kids couldn't operate the toy detector themselves, for obvious reasons. Had they been able to do that, I'm sure we'd have been running full tilt the whole time!