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Open Lockers

 

 

Hundred Lockers is an exercise about number properties and parity.

The animation above shows the state of the lockers after each passage of Willy Wonka’s assistant.

Image [100] provides the solution.

The answer is the square numbers: 1,4,9,16,25,36,49,64,81,100.

At the end of the procedure the status of the locker (closed 0, open 1) is  the number of factors of locker number N mod 2, e.g.

  • locker 10: 10 has 4 factors (1,2,5,10), 4 mod 2 =0, hence the locker is closed
  • locker 25: 25 has 3 factors  (1,5,25) 3 mod 2 =1, hence the locker is open

Square numbers have an odd number of factors:

Every number can be expressed as the product of power of primes:

N =\prod_{j}^n p_j^{j_k}

Let \phi (N) be the number of factors of N:

\phi (N) =\prod_j^n (j_k +1)

For square numbers all j_k are even, hence \phi (N) \text{ mod } 2 \equiv 1

 

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Categories: Mathematica, Mathematics
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