More Dots:

Try Sim starting 7 or 8 dots.

One-Color Sim:

Both players draw in the same color. The first to make a triangle (with all three corners on dots) loses.

Triangles in Triangles:

Make a triangle with 6, 10, or 15 dots.

In this variation, both players use the same color. Take turns connecting a pair of neighboring dots. The first player to connect a pair of dots that completes a triangle of any size loses.

Sides and Diagonals:

Say you've got 6 dots arranged in a circle, like at the start of Sim. These dots can be connected to make a hexagon.

Each of these lines connects a pair of neighboring dots. These are called, unsurprisingly, the "sides" of the hexagon. Lines that connect non-neighboring dots are called the "diagonals" of the hexagon.

A hexagon has 6 sides and 9 diagonals. Draw 5 dots in a circle. Can your child figure out the number of sides and the number of diagonals? What about for 7 dots, or 8? Don't give away the answers, but in case you need to check your work: 5 dots form a 5-sided shape with 5 diagonals (a pentagon), 7 dots form a 7-sided shape with 14 diagonals (a heptagon), and 8 dots form an an 8-sided shape with 20 diagonals (an octagon).

Ties:

Can Sim end in a tie? In other words, is it possible for both players to run out of moves without anyone making a triangle in their color? Give it a try! See if, using two colors, you can fill in a game of Sim with no triangles of a single color. Don't look ahead at the next two Learning Notes until you've tried.

5-Dot Draw:

Okay, have you tried drawing a Sim game with no ties? If not, go back to the last bullet point and give it a try before reading ahead! The answer is... 6-dot Sim can't end in a tie. When all the lines are drawn using two colors, there will always be at least one triangle of a single color. But, 5-dot Sim can end in a tie. Can your child draw a tie-game of 5-dot Sim? In other words, challenge your math beast to connect every pair of dots in a pentagon, using two colors, without creating a triangle with all sides of the same color (and with corners on the original dots).

14 Lines, No Triangle:

Regular 6-dot Sim can't end in a tie, which means when all 15 lines are drawn in there must be at least one triangle with all sides of the same color. But, it's possible to draw 14 lines in a Sim game, using two colors, without making a single losing triangle (that is, without drawing a triangle with all sides the same color). Can your child (or you) draw these 14 lines? One answer is shown below. Don't peek if you haven't tried yet!