## CNB3 Neck-tie Examples

In CNB2 I showed you the Neck-tie diagram, and described how it is used to construct the rows of the cycle-number triangle T, for rows R1, R2, R3, etc., and asked you to figure-out from my garbled explanation how it is done. To make the matter clear, here are the neck-ties for these first three rows. Mark carefully what they look like and where they come from, and then sketch the next three for yourselves. I use the symbol **ni** for the *i*th neck-tie, with *i* = 1,2,3, … Of course, we cannot have an **n0** neck-tie, because it couldn’t have a neck triangle!

**Neck-tie n1** for row **R1** ; the cycle-number **1** = **111 …** You can take this from either the right-leg of the neck-tie, or the left-leg, cycling the 1 downwards.

**Neck-tie n2** for row **R2** ; the cycle-number **2** = **101010 …** starts from the second element **1** in **R2** , then it takes the corner element **0** , making the fundamental cycle **10′** , then it continues cycling down the right-leg towards infinity.

**Neck-tie n3** for row **R3 ; **the cycle-number **3**** = 110110** … starts from the second element **1** in **R3** , then it takes the corner element **0** , making the fundamental cycle **110′** , then it continues cycling down the right-leg towards infinity.

Having made the concept of ‘neck-tie’ clear to you, I hope you will buy a notebook with pages printed with cells in a grid (as shown in my diagrams; usually called a quad book) : and construct a **CNT** down to, say **T(20)**. You should not show the neck-ties, which were only introduced to define the procedure for row-to-row generation of **T**, and to help define ‘cycle-number’. Of course, you may omit the grid-lines too. That is easy to do in a print-out, if you have produced your triangle on your PC, perhaps using microsoft WORD (TABLE) or EXCEL.

As you draw your triangle, you will immediately see patterns in the rows and diagonals (and columns) of **T**.

Two questions, to finish:

Have you tried adding the 1’s in each row? Does the column of sums mean anything to you?

How should we define ‘prime cycle-numbers’?