(For a related post see

my post on Schur's theorem. The paper THIS post refers to is

here.)

(ADDED LATER: A commenter pointed to Graham-Knuth-Patashnik for a closed form for pennies, nickels, dimes, quarters, half-dollars. I have wriitten up their account and it is available

here. I also apply their technique to just get 1,5,10,25.)

How many ways are there to make change of a dollar using pennies, nickels, dimes, and quarters? This is a well known question; however, the answers I found were disappointing. They were of two types

- There are 242 ways to make change. The author then point to a program he wrote or to the actual list of ways to do it.
- The number of ways to make n cents change is the coeff of x^n in the power series for 1/(1-x)(1-x^5)(1-x^{10})(1-x^{25}). This can be worked out. I have never seen it worked out.

The first answer yields an actual number but is not interesting mathematically. The second answer is interesting mathematically but does not easily yield an actual number.

I looked for a closed form on the web but could not find one. So I did it myself. The paper is pointed to above. I did not use generating functions, just recurrences. I do not know if it is new. Whether it is old or new I hope you enjoy it. The paper actually gives a closed form for the coin sets {1,x,kx} and {1,x,kx,rx}. Along the way we derive the answer for a dollar and the usual currency by hand.

This raises some questions:

- Is my formula new? I'd be surprised either way. (Is it possible to be surprised at both statement A and statement NOT(A)?). If it's new I'll be surprised since the questions has been around for so long and surely someone would have done it. If it's known I'll be surprised since (a) I went to JSTOR and searched for all math journals they had for the words ``change'' and ``coin'', looked at the over 400 such articles (just the titles and abstracts) and didn't find it, (b) this came up in math.stackexchange here and, true to form, every answer was either a program or a generating function and (c) other searches also did not turn up the result.(ADDED LATER- Since the 1,5,10,25,50 case was known, I guess I AM surprised.)
- Even if its not new it is clearly not well known. Why is that? Its a natural problem that people seemed to be interested in. One conjecture: The COMP SCI people interested in it were happy to write a program. The MATH people interested in it were happy to say `its merely the nth coeff of...' So a closed form solution seems to not be of interest to either camp
- Is it interesting? (a) Comp Sci: the closed form solution gives an answer in O(1) steps instead of the usual dynamic programming O(n) solution, and (b) Math: the closed form solution tells you the coefficients of the power series of 1/((1-x)(a-x^5)(1-x^{10})(1-x^{25}).

Side Bar: I asked the 10 students in my REU program to just GUESS how many ways there are to make change of a dollar with pennies, nickels, dimes, and quarters. Most made guesses under 100. The lowest guess was 30. One person guessed 100! and one guessed 100!/2. I then told them that it was between 30 and 100! and assigned them to derive it themselves by hand. Most were able to.