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geodave

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Well, I'm partial to squares (4, 9, etc.) because of the open locker problem*, among other reasons.

 

Also, DNA is base 4, so it's sort of built-in. :P

 

* Take 100 school lockers (numbered 1-100). Open every one. Close every second one. Toggle (open or close) every third one, etc. Which lockers are left open at the end?

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Any defenders for the poor base-3 system?

 

Problem: Break a 40 kg weight into 4 pieces such that those pieces can then be used to measure any (integer) weight from 1 to 40 kg using a balance.

Solution uses balanced ternary (digits: -1, 0, +1).

 

- Madhav.

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What exactly do you mean by "etc."?

 

Hm, do you stop after 50? Or continue til 100?

 

Solution uses balanced ternary (digits: -1, 0, +1).

 

What exactly do you mean by that? (I have no clue.)

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Given he was talking about squares, I assume he goes up to 100. If the problem were to continue "for every number n from 3 to 100,, go through and toggle every locker that is a multiple of n", then the lockers open at the end would be lockers which are squares, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

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What exactly do you mean by that? (I have no clue.)

 

Balanced ternary <3 (there's actually a level in my upcoming set with it as a theme...lol)

 

It's base 3, but the digits are -1, 0, 1 instead of 0, 1, 2.

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Hm, do you stop after 50? Or continue til 100?

 

If I remember the problem correctly, you keep going until 100.

 

 

I have no idea what the heck base systems are (Y)

 

You're not missing out on much. :teeth:

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I have no idea what the heck base systems are (Y)

 

It's not hard. We use the decimal or base 10 system. In this system there are 10 digits, 0 through 9, representing each number less than 10. Each number N is represented by a list of digits, which are determined by finding d_1, d_2 etc. such that N = d_1 + 10*d_2 + 100*d_3 + 1000*d_4 + ... and so on until we reach a power of 10 larger than N. We write numbers using this series of digits in opposite order from how I wrote them there.

 

Ex: 1028 is 1*1000 + 0*100 + 2*10 + 8

 

In an alternative base system it's the same thing, only 10 is replaced by some other number, and there are a different number of digits, which may be represented by some smaller set of numbers or it might have 0-9 plus extra symbols if the base is larger than 10.

 

Ex: In base 2, one has the digits 0 and 1 representing the numbers less than 2. A number can be represented in binary with digits d_1, d_2 etc. such that N = d_1 + 2*d_2 + 4*d_3 + 8*d_4 + ... and so on until we reach a power of 2 greater than N. Remember that these d's are required to be 0 or 1, they are not allowed to be anything else.

 

234 in binary, for example, would be 1*128 + 1*64 + 1*32 + 0*16 + 1*8 + 0*4 + 1*2 + 0*1, so we would write it as 11101010.

 

The base 2 system is very popular because it is used by computers to represent numbers, because computers store and operate on data as a series of on/off states, which are made to represent numbers in binary.

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Yes, the answer to the locker problem is all the square numbers. What this means, really, is that square numbers are the numbers that have duplicate divisors, and so end up with an odd number of toggles.

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Pretty much. We used to love fooling around with those at college :P

 

Also, I remember solving that problem last semester in my class :P

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