## How see-saws should be made April 13, 2007

Posted by dorigo in games, mathematics, personal, science.

A see-saw is one of the toys you frequently find in children parks and anywhere small children need outdoor entertainment. It is usually built as a long log of  wood with a support in the middle and seating for the kids on both ends.

I was looking at my kids playing on one such thing last week, while enjoying a hot sun at the Rifugio Gallo Cedrone, (2200 m above sea level), in val Pusteria, and it just dawned on me that those things are not built the way they should.

My intuition told me that the see-saw should not be perfectly balanced when unloaded, but biased in advance, if the aim is to provide the maximum number of pairs of kids with perfect fun. In fact, when you play with a see-saw and your playmate is heavier or lighter than you, there is arguably less fun, since more effort is needed to perform those ups and downs. A biased see-saw should allow more odd pairs to enjoy the game.

I had forgotten the matter, but yesterday I read a post on http://kea-monad.blogspot.com/ where Kea discusses a simple geometric problem, totally unrelated to the one of designing a see-saw. However, both puzzles belong to a whole class of mathematical problems whose solution involves finding a minimum in a simple function. So I decided I would demonstrate that my intuition was correct.

Today, armed with paper and pencil and ten minutes of free time, I did solve the problem. You want to minimize the squared difference of total weight on the two ends of the see-saw by applying a bias x, assuming two kids with weight w1 and w2 (and posing w1>w2) are sitting with w2 on the side of x. That is, you want to find the value x which corresponds to a minimum of f(x) =  (x+w2-w1)^2.

Further assume that w1 and w2 come from a flat distribution of minimum m and maximum M. This may not correspond to the reality of pairs of kids in a playground, but it is a reasonable ansatz – which can be changed anyway. If you do that, the problem reduces to that of a simple integration on w2 from m to w1 of (x+w2-w1), followed by another integration on the result with w1 variable from m to M.

Confused ? It is a problem for junior high school students. The solution is simple: the best value of the bias x is just (M-m)/3. That is, if you assume kids will play on the see-saw if they have a weight between 20 and 80 pounds, the see-saw would do them a better service if it came biased by 20 pounds on one side!

1. marco - April 13, 2007

Interesting thought!
Now you should really sample 100 kids or so at the playground, weight them, build the see-saw accordingly and trying it out on another 100 kids.
Just for the sake of experimental verification, you know 🙂

2. Kea - April 13, 2007

Tommaso, I can see that your kids are going to grow up to be very good physicists. You are quite right that ‘symmetry breaking’ is something a child could understand (as Louise would say). Have a great day!

3. Carl Brannen - April 14, 2007

The seesaws we used were sufficiently primitive that one merely put the heavier person closer to the fulcrum.

4. Fred - April 14, 2007

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My favorite game was tetherball (with dodgeball and ‘guardie e ladri’ a close second), though I’m glad I didn’t have to figure out the math for it. If you pounded the ball in a particular way, you could create some nicely distorted paterns as the ball and rope danced while revolving around the pole.

5. Markk - April 15, 2007

I agree with Carl. Why do you want the heavy kids always to have to sit on the same side of the seesaw? Thats the crummy side (as the kids would say). Much easier to just have the heavy kid sit a little closer in. If the the difference in mass is too great the bigger kid (often a parent) just controls the up and down anyway.

6. Kea - April 15, 2007

Re Carl’s idea: all well and good in the old days, when planks were used to make see-saws. Nowadays (at least around here) they use metal poles, which would be rather uncomfortable for the tubby ones.

7. Kea - April 15, 2007

Tommaso! We can invent a shifting-bias see-saw, where the extra weight can sit on either side of the centre depending on the whim of the previous see-saw users.

8. dorigo - April 15, 2007

Hi folks,

good idea Marco… If I was a designer of toys I would take it in consideration.

Carl, of course there are several ways about this theoretical problem – even a uneven length would do. In fact I remember seeing a see-saw with two seats on each side, so up to four kids could play. And if only two were there, they could take odd seats for a better balance.

Fred, what is the meaning of that letter arrangement, besides my name in reverse ? An acrostic maybe ?

Kea, you are right – in a sense this is a very simple case of broken symmetry. And one could certainly design a shifting-weight see-saw, but I guess it would look a bit like overdesign!

Cheers,
T.

9. Fred - April 15, 2007

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Your name as a see-saw but your system collapsed it so maybe a wall on the far left side will fix it.

10. Fred - April 16, 2007

Nope.

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Only with stars, of course.

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