Let’s imagine for a moment a game of Infinite Tak, with infinitely many pieces and played on an infinite board. What would happen? For starters, such a game would never end — the end conditions rely on either a finite board or finite pieces. Also, black starts at a comparably minuscule disadvantage as he literally has all the time in the world to catch up; indeed, this is true for either player when they fall behind in any way. But the lack of an achievable victory doesn’t makes this analysis futile, we just need to redefine the objective. The goals of this game can be stated fairly simply. At any given time, you are trying to: 1) have the longest connected path, and 2) have the highest flat count. Note that in order to match the finite game’s rules, the “length” of a path isn’t its Euclidean distance, but rather its width or its height (whichever is greater). In pursuit of these goals — or perhaps just out of sheer boredom — some interesting patterns will eventually emerge, and it’s an intuition for these patterns that I want to build. In this open-ended series of posts, I will be introducing various patterns as though they have arisen in the context of Infinite Tak, and then selectively delve into how this understanding extends to the finite game.

Beyond a solo flat, the most basic pattern you can make is a partnership — two touching pieces. A partnership is a fully connected shape and may thus add value to your longest connected path immediately. This is the most cooperative pattern of two pieces because they directly support one another. Although the partners only extend in one direction, they also provide orthogonal redundancy as a tempo-neutral recapture and by creating an extra lane for expansion. In fact, this redundancy is arguably more valuable than the extension, which stubbornly commits to a single lane and loses directional tempo when recapturing.

Partners are well suited to hotly-contested regions, where building local influence is vital. And if you can correctly anticipate competition in an area, adding partners there preemptively will be a fantastic asset. However, partners can also be an over-commitment of resources; in less important areas a looser shape might be more appropriate. Since they expand your influence so slowly, you only want to add partners away from the action when you have time to “press the initiative”. There are two common scenarios when this will be the case: 1) you have thoroughly dismantled a segment of your opponent’s path and want to build out your own, and 2) when you have thoroughly dominated an area of contention and want to increase the value of that victory by supportively extending away from the area.

Getting away from Infinite Tak, we find that edges introduce some caveats to the partner shape. Partners that both connect to the edge are a basic form of “edge-crawl”. This is not a particularly strong shape, offsetting the orthogonal redundancy benefits by foregoing potential influence. Your opponent can seek orthogonal partners in the path of your extension, which provides a relative benefit to both their tempo and influence. And if you block their edge partner, it creates a wall-able hole in your edge-crawl that will either devastate your tempo or draw your capstone out of position.

In contrast to edge-crawling, it is actually quite valuable to give an existing edge piece a non-edge partner. Attachments to the edge are important because they are necessary to actually finish a road, and so you should prize your pre-existing edge pieces — having to re-attach to the edge can sometimes be tricky. Indeed, a preemptive center partner is invaluable because it will never need to recapture the edge; if your opponent develops an edge-bound attack, then you can back up the edge piece with another partner. Note that this is a unique feature of edge positions specifically because they have an odd number of neighboring positions.

Partners near the edge can also be quite powerful. Although partners add an additional lane for orthogonal expansion, this strategic influence decays the farther away you get. In other words, once you start to develop orthogonally off of partners your path starts to look much narrower; alternately, your opponent can still effectively get in the way of both lanes if they start their defense far enough away from you. Thus, partners near the edge are an excellent way for ensuring an eventual attachment to the edge because their dual-lane flexibility has not decayed.

Back to our more abstract Infinite Tak reasoning, capstones are obviously an excellent partner for any solo flat. Consider the ultimate objective of the partners. If they are being created for extension, then the capstone is liable to remain a passive anchor for your position. If they are being used more for flexibility reasons, then you are anticipating a much more active cap whose function will be to reclaim captured segments of your road. As always, capstones are best played where the action is thickest, but they do not necessarily need to be active in order to achieve value — they will continue to pay off as long as they remain relevant.