One of my daughter’s favorite toys is this puzzle set that forms a track:

At only six months she’s not old enough to set up the track, but she loves to watch the mechanical seal that travels whatever path is laid out for it.

As the seal passes certain points, it makes predetermined sounds that go with the picture. For example, passing by the dolphins causes dolphin sounds to occur.

What’s interesting mathematically is how the track manages to “code” the different sounds. Notice there are raised ridges on the track. There are four ridge positions matching with four switches on the back on the seal.

If “raised” is a 1 and “lowered” is a 0, then the dolphin picture shows the code 1010.

Hence each sound is linked to a certain combination of switches. What’s particularly interesting is that the track has to work in both directions (it’s not preset which way the seal is facing) so that the switch combinations have to either be palindromes (like 0110 backwards is still 0110) or indicate the same sound when interpreted backwards (so 0101 will give the same sounds as 1010).

Hence, the maximum number of sounds possible on the track given four switch positions is nine: 1001, 0100, 0110, 1000, 1100, 1010, 1110, 1101 and 1111. (Any of the codes may be reversed.)

The actual track uses five: 1001 (crab), 0110 (seal), 1010 (dolphins), 0100 (scuba diver), and 1100 (tugboat). I believe the codes involving three and four 1s are omitted due to the device being slightly imprecise (as is once in a while it gives the wrong sound) and it being hard to hit 3 or 4 switches consistently.

All this leads to the following question, which is reasonable for a high school level: **Given a track system with ***n* switches, how many possible sounds can it make?

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Rick Regan, on September 25, 2009 at 8:16 am said:Pretty cool. I want to buy one just to see it firsthand.

I agree there are nine (since 0 doesn’t work), but in your list of nine possible sounds you list 1001 twice. You should add the pair 0001/1000.

BTW, you say the “actual track uses six” but you only list five. (If there are six I’d expect the sixth to be 0001/1000.)

Jason Dyer, on September 25, 2009 at 5:40 pm said:Blimey, you’re right.

EDIT:Fixed. Extra 1 was a typo as was six instead of five.Rick Regan, on September 26, 2009 at 8:53 am said:As for your question “Given a track system with n switches, how many possible sounds can it make?”, I came up with two formulas:

– When n is odd: 2^floor(n/2) + 2^(n-1) – 1

– When n is even: 2^floor(n/2)/2 + 2^(n-1) – 1

I’ll skip the derivation here, but the idea is to count the “palindromes” (I put that in quotes because palindromes don’t normally include leading 0s), count the others by dividing them by 2, the subtracting 1 since 0 is invalid.

(I’m not sure that was high school level.)

Jason Dyer, on September 26, 2009 at 11:00 am said:My colleague’s Contemporary Math class (with normal students) features problems of roughly the same difficulty, and I’d feel comfortable giving this problem to my Honors Algebra II class.

Walking Randomly » Carnival of Maths #58, on September 27, 2009 at 1:35 pm said:[…] up, we have a post from Jason Dyer of Number Warrior where he analyses one of his baby daughter’s favourite toys. That’s right ladies and gentlemen, only 6 months old and already inspiring […]