I am trying to solve for possible combinations of weights that would be appropriate for use in my home gym. I have been told that this is a case of integer partitioning, but I am not sure how to solve it and would appreciate some guidance (or even better a solution!).
Given that I am quarantined at home, I want to buy weight plates for use in a home gym. In my gym, I have a 45lb bar. I attach weight plates to the bar in pairs, so I always purchase and use plates in pairs. So, if I buy 1 pair of 2.5lb plates, I would be able to use the bar at weights of 45lb or 50lb (since plates must always be used in pairs).
I want to determine how many pairs of 2.5lb, 5lb, 10lb, 25lb, and 45lb plates I need to ensure that I can use my bar at weights of 45 to 405lb in 5lb increments (so options would include 45, 50, 55, 60,..., 395,400, 405). I want to minimize the number of smaller plates that I buy (so mostly get 45s and then get some minimum number of pairs of each of the smaller plates to ensure all possible combinations are attainable).
How might I approach this problem?