How Not To Feel Pain At The Dentist

Before my last visit to the dentist, I wanted to know the number of the tooth that was hurting. I found out that there was a certain Universal Numbering System . Great! When I got there, I told him that number 3 was hurting. Well, it happens that the Universal system is not so universal. They told me that tooth was number 16. I was confused, because I thought that if it were number '16', the maximum would be 18 since there are 2 teeth after it, and 18 x 2 = 38, but we only have 32 teeth. So I knew something was wrong with the notation. In fact, they told me that this "Universal" system is used in the U.S. but the rest of the world uses the FDI World Dental Federation notation.

"Universal":

Permanent Teeth
upper left								upper right
16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
lower left								lower right

FDI:

upper right								upper left
18	17	16	15	14	13	12	11	21	22	23	24	25	26	27	28
48	47	46	45	44	43	42	41	31	32	33	34	35	36	37	38
lower right								lower left

(As you can see, right and left are mirrored in the tables)

Since I like Math, I began to search for an injective function from the Universal number to the FDI. So let's begin.

The FDI divides the teeth in quadrants. The quadrant may be given by



So we multiply it by 10 to shift the number left



Sometimes the numbers increase in the quadrant, other times they decrease. So we need to alternate by quadrant:



For odd quadrants let's pick the constant 8. And for even quadrants, the constant zero:



Now let's have something varying up and down:



Now to get the unit in the FDI system, from 1 to 8, we use the expression:



So, for the final formula we have:



Next time you go to the dentist, just try to reconstruct the formula to take your mind off the pain.