20150916 HOLES AS TOKENS:

A proven method to make them very mad.

If you don't see the picture, please visit my website defmath.com. Google does not like vector graphics.

The shnumber on the board is equal to -30814. How? I made it out of holes, or empty spaces, and turned upside down.

Telling such shnumber is relatively easy. Making operations with holes requires silencing your intuition and following the rules. This is quintessential for mathematics. Just try.

Besides being a brain twister, such shnumbers are models of negative quantities. Male elements are ones, female elements are minus ones (please excuse me for being an electronics engineer). When the hole is filled, it becomes zero.

This model once made TLG's little sister very frustrated. I did not offer it to TLG yet.

Putting a positive board on top of a negative board, we can add up negative and positive quantities. Fortunately, there are other ways to do it, because filling boards with pegs is taxing. Unfortunately, I can't offer or recommend a computer program for such manipulations. You can drag and snap circles using any free object-oriented editor, don't even think of letting kids do it though, because of the presence of the shape-changing controls.

Remark that adding a positive number to a negative number you perform subtraction (with "borrowing"), but nothing prevents you from subtracting bigger number from smaller number. You simply end up with unfilled holes.