3 Number Combination Lock Possibilities

Extensions for the Simplex Lock Problem

1 Any Number of Buttons

One way to extend the Simplex Lock Problem is to consider not only a 5-push lock, but locks with any number of buttons. If you accept an n-button Simplex Lock, how many combinations will it have?

2 Picking a Lock

The five-push Simplex Lock has only 1,082 possible combinations. By comparison, this 3-dial lock (three wheels, each with digits 0-9) has x � 10 � 10 = i, 000 possible combinations.

The total number of combinations is not very dissimilar, just the Simplex Lock is much harder to choice because it is harder to systematically test each possible combination.

In the lock in a higher place, you can simply listing numbers 000-999, lowest to highest, and you will test each combination. If you modify the lock to 4 dials, the number of combinations is now 10,000, but you can all the same step through all of them using the same strategy: Listing the numbers 0000-9999.

How could y'all systematically test each combination of a 5-button Simplex Lock? This requires finding a way to listing them all without missing whatsoever and without duplication. Can you lot extend your algorithm for lisiting combinations to an n-button lock?

3 Stirling Numbers

A student in Oakland, California, came up with the following idea: You brand a picture like Pascal'south triangle, except that in adding entries from 1 row to the next, each number on row N has a weight, which is its position within the row (starting from 1). For example:

1[1] 1[1] 1[2] 1[1] 3[2] 2[3] 1[1] 7[2] 12[3] 6[4] 1[1] 15[2] 50[3] 60[4] 24[5]

The numbers in brackets are the weights. Then for example the 7 in the fourth row is 1 � ane + 3 �2. The l in the fifth row is 7 �2 + 12 �3. And the sum of each row is (starting at n = 0) is the number of combinations in ann-button Simplex lock."

The numbers in this triangle are related to the famous "Stirling numbers of the second kind". Find out nigh Stirling numbers and use them to discover a solution to the Simplex lock trouble.