Tag Archives: pentago

Pentago, Geometry Daily, and The OEIS

Welcome to this week’s Math Munch!

Pentago Board

Hurricane Sandy is currently slamming the East coast, but the Math Munch Team is safe and sound, so the math must go on.  First up, if you’ve visited our games page lately, you may have noticed a recent addition.  Pentago is a 2-player strategy with simple rules and an enticing twist.

  • Rules: Take turns playing stones.  The first person to get 5 in a row wins.  (5 is the “pent” part.)
  • Twist: After you place a stone you must spin one of the 4 blocks.  This makes things very interesting.

Why don’t you play a few games before you read on?  You can play the computer on their website, play with a friend by email, or download the Pentago iPhone app.  But if you’re ready, let’s dig into some Pentago strategy and analysis.

Mindtwister CEO, Monica Lucas

Mindtwister (the company that sells Pentago) put out a free strategy guide that names 4 different kinds of winning lines and rates their relative strengths.  The weakest strategy is called Monica’s Five, and it’s named after Mindtwister CEO and Pentago lover, Monica Lucas.  You can read our Q&A for more expert game strategies and insights.  We also had a chance to speak with Tomas Floden, the inventor of Pentago, so it’s a double Q&A week.

As you play, you start to build your own strategy guide, so let me share three basic rules from mine.  I call them the first 3 Pentago Theorems.  (A theorem is a proven math fact.)

  1. If you have a move to win, take it!  This one is obvious, but you’ll see why I include it.
  2. If your opponent is only missing one stone from a line of 5 you must play there.  It seems like you could play somewhere else and spin the line apart, but your opponent can play the stone and spin back!  The only exception to this rule is rule 1.  If you can win, just do that!
  3. 4 in a row, with both ends open will (almost always) win.  This is a classic double trap.  Either end will finish the winning line, so by rule 2 both must be filled, but this is impossible.  The exceptions of course will come when your opponent is able to win right away, so you still have to pay close attention.

Up next, check out the beautiful math art of Geometry Daily.

#288 Fundamental

#132 Eight Squares

#259 Dudeney’s Dissection

#296 Downpour

#236 Nova

#124 Cuboctahedron

#136 Tesseract

#26 Pentaflower

#92 Circular Spring

The site is the playground for the geometrical ideas of Tilman Zitzmann, a German designer and teacher, who’s been creating a new image every day for almost a year now!  He also took some time to write about his creative process, so if you’re interested, have a read.  Visit the Geometry Daily archives to view all the images.

Finally, an amazing resource – the On-Line Encyclopedia of Integer Sequences.  What’s the pattern here?  1, 3, 6, 10, 15, 21, …  Any idea?  Do you know what the 50th number would be?  Well if you type this sequence into the OEIS, it’ll tell you every known sequence that matches.  Here’s what you get in this case.  These are the “triangular numbers,” also the number of edges in a complete graph.  It also tells you formula for the sequence:

  • a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+…+n.

If you make n=1, then you get 1.  If n=2, then you get 3.  If n=5, you get the 5th number, so to get the 50th number in the sequence, we just make n=50 in the formula.  n(n+1)/2 becomes 50(50+1)/2 = 1275.  Nifty.  Who’s got a pattern that needs investigating?

Have a great week, and bon appetit!