Batch 2 - Class 314 - Celtic Knots

• What if you wanted the picture to fall if you remove 2 out of n nails? (See the advanced paper in links below). Solve for 3 nails

• Celtic Counting requires students to trace an under and over pattern – counting the Celtic loops. How many loops exist in this knot?

• Answer: Two
• How many loops does this knot have?

• Answer: Three
• Instructor Notes: Let kids try out a few other knots themselves
• Square Knot Patterns
• How many loops does the following knot pattern have?

• Answer: Six - each a rectangle
• Lets take a 6x7 knot - how many loops does this have?

• Answer: One!
• Lets take a 6x8 knot - how many loops does this have?

• Answer: Two
• How about a 6x9 knot?

• Answer: Three
• Any idea on what is going on? Can you try and guess the formula for square knots?
• Do you notice the symmetry? Following is the "center loop" and the two fishes.      

• What can you say about symmetry in a nxm knot?
• Instructor Notes: Some kids may come up with notion of Greatest Common Divisor. Else motivate and take other examples like 6x12 or 5x10. Advanced kids may get the intuition and understanding
• Why does the GCD determine number of loops?
• For pxq knot, LCM(p,q) is where the knot will retrace itself. This represents pj=qk=LCM for some j and k. Number of loops is p/k (or q/j) which is GCD since GCD.LCM=pq
• Can also think of this as a pattern of the full knot repeated in a grid. So instead of bouncing off an edge, continue over on block. One will repeat on a loop after LCM/p heights or LCM/q widths, and by then the loop/straight line would have covered p/GCD points on left edge and q/GCD points on top edge of original grid. Since all p left-points have to be covered eventually, the number of loops is GCD.
• Solve some of the more advanced ones on the exercise sheet
• Draw your own loops! and solve!

• What is p and q in each of the above
• The first one is 2.5 x 12, the second is 2 x 12 - note that the value of p can be a half integer here. Why?
• Count the number of loops in each. Is there a pattern?
• Can you explain this pattern?
• Number of components is GCD (2p, q) - the argument is the same as before. We now have to traverse vertically twice for each horizontal traversal