Prime Number Cryptarithms

Solutions are below:

The Balloon Problem

Solution below:

Paper Folding Problem

Solution below:

Untangling Loops of Yarn

Solution below:

The easiest way to solve these puzzles is to check the loops one pair at a time. 

TOP LEFT:  Notice that for the loops of yarn in this puzzle, pink and green are not linked, but green and blue are.  Pink and blue are also linked.  If we cut the blue loop, therefore, the loops will all come apart.

TOP RIGHT:  All 3 loops are each connected to each of other, so you would have to cut any two loops for them all to come apart.

BOTTOM LEFT: Only blue and green are linked, and so we can cut either one for the loops all to come apart.

BOTTOM RIGHT:  Pink is linked with the other colors, but blue and green are not linked.  So we should cut the pink loop.

Challenge:  The chart below represents a giant tangled ball of yarn loops.  It shows which pairs are linked to each other.  What is the smallest number of cuts needed to separate all 7 colors?  Notice, for example, that if you cut red and pink, for example, then you don't need to cut orange because orange would already fall loose.