Another week, another riddler. This one is from Austin Shapiro:

*From Austin Shapiro comes a story of stacking that may stump you:*

*Mira has a toy with five rings of different diameters and a tapered column. Each ring has a “correct” position on the column, from the largest ring that fits snugly at the bottom to the smallest ring that fits snugly at the top.*

*Each ring she places will slide down to its correct position, if possible. Otherwise, it will rest on what was previously the topmost ring.*

*For example, if Mira stacks the smallest ring first, then she cannot stack any more rings on top. But if she stacks the second-smallest ring first, then she can stack any one of the remaining four rings above it, after which she cannot stack any more rings.*

*Here are a four different stacks Mira could make:*

*This got Mira thinking. How many unique stacks can she create using at least one ring?*

*Extra credit: Instead of five rings, suppose the toy has N rings. Now how many unique stacks can Mira create?*

To find this we brute forced solutions for rings of 3 through 7. Here are visualizations for the five and six ring solutions: