Why the Same Key can Open Different Padlocks?

I have experienced that one key can be used to open different padlocks. The padlock keys are now very simple. They are made of square-shaped alloy which is then ground at different sizes and positions to distinguish one key from another. Because the key length is limited and the size of the ground part is also limited, there is a maximum number of different keys that can be made. What is such a maximum number? Let us use statistical physics to estimate it [1].

The part of the key, both ground and unground, can be considered as a collection of spins. See Figure 1 as an illustration. The ground parts are made of a collection of up spins and the unground parts are made of a collection of down spins. Suppose the total length of the functional part of the key is L, the total length of the unground parts is P, and the total length of the ground parts is Q. Suppose the minimal length than can be differentiated by the grinding machine (grinding machine accuracy) is a. Thus, the corresponding total number of spin, total number of up spin, and the total number of down spins are

respectively. The number of different keys composed of P unground length and Q ground length that can be made becomes

Suppose the minimum length that can be differentiated by the grinding machine is a = 1 mm. Suppose also the length of the functional part of the key is L = 1.5 cm = 15 mm (N = 15). The number of different key having the ground part of Q = 0.6 cm = 6 mm (n = 6) and P = 0.9 cm = 9 mm (m = 9) is w(15,6) = 15!/6!9! = 24,226 different keys.

Since the number of either ground and unground parts varies from zero to L, then the total number of the different keys at any values of the ground and unground parts is

As an example, for N = 15 we obtain W(15) = 32,823 different keys. We have used the Wolframalpha [2] to calculate the summation. Therefore, if the number of the same size padlock keys available in the market is larger than 32,823, the identical keys must exist.

References

[1] Mikrajuddin Abdullah, Mekanika Statistik, Bandung: Penerbit FMIPA ITB (2017).

[2] https://www.wolframalpha.com/

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