3-3 invasion  

• b, 4-4 Point 3-3 Invasion, Hane-Extend, Push - (joseki) (beginner) (AI favorite)
• t, 4-4 Point 3-3 Invasion, Hane-Extend, Tenuki - (joseki) (intermediate) (AI favorite)
• a, 4-4 Point 3-3 Invasion, Hane-Extend, Hane - (joseki) (common) (beginner) (traditional)
• c, 4-4 Point 3-3 Invasion, Hane-Extend, 5-Space High Pincer - (rare)

Overview of Joseki

Territorial Play

Hane-Connect  

The traditional joseki continues with hane and connect. This guarantees life in a sizable corner, but Black is extremely thick and the tiger's mouth projects a large amount of influence. Historically, it was believed that Black is better in this type of situation, as the value of Black's influence is greater than the tangible value of White's territory in the corner.

This belief is still true in the AI era, but only if White plays the hane. Many professionals describe the hane as a mistake that makes Black too thick. If White omits the hane, the 3-3 invasion is an even result because Black's wall is still attackable.

It is important to note that these recommendations are relevant in the context of an empty board in the early opening. Depending on the board position and situational context, the hane and connect can still be a good territorial move. In fact, some professionals choose to tenuki after because they plan to await more information before deciding to attack or take territory.

Attacking Play

Push and Pincer  

Ever since the AI revolution, professionals and strong AI programs recommend crawling once with to guarantee life and then tenuki. The purpose of playing this way is to preserve the possibility of attacking Black's wall in the future. Black's wall is influential but not yet thick: it is lacking eyeshape. In fact, it is very common for amateur players to immediately pincer around x and/or y, which later threatens to take a peep at z.

Some players would say that Black's wall isn't thickness, but rather a target. The moves at y and z have a convenient geometric relationship (large knight) that is efficient play for White. Regardless of the outcome, it is easy for White to develop a stable position on the upper side of the board, which makes it difficult for Black to make good use of their wall. In many circumstances, Black can end up over-concentrated.

Crawling with on the second line is essential for any kind of pincer strategy. Without , almost any move that Black takes against the corner is sente, which would negate an attempted pincer attack. Although crawling on the second line is traditionally called "the line of defeat", it is a necessary move that paints a target on Black's wall.

Tenuki

Mistake  

A common beginner mistake is to hane with instead of extending. This results in a collapse of Black's position due to multiple cutting points.

This position is sensitive to ladders, but White can obtain a good result regardless if the ladder works or not.

For replies to this mistake see 3-3 Point Invasion and RTGProblem31.

When should white exchange the hane and tiger's mouth?

Shi Jingyao? vs. Yang Bowei (2020)  

The hane and connect can still be an AI/professional recommended move in certain circumstances.

When the 3-3 invasion is played in the middle game or later (which is rare in modern go), sometimes Black may have added thickness on the outside that impacts our assessment of the corner. In Shi Jingyao? vs. Yang Bowei (2020), Black has a double ponnuki that makes them extremely strong in the center of the board. It is unlikely that any pincer strategy will prove fruitful for White, as this would be like jumping into a den of tigers. As the proverb goes, White should play away from thickness.

White has no plan to pincer Black's wall, so the hane and connect is a territorial move that maximizes the size of White's corner. Moreover, White is not concerned about making Black even thicker in an area where Black is already strong. While it does seem like Black will obtain a large moyo, White has already calculated they have enough points to win the game. The hane is a calm move that White plays to say that they are winning, and there isn't a need to risk a fight.

In short, the hane can be a reasonable move when Black is already thick or overconcentrated.

Should white push or tenuki?

References

• [1] Frequency statistics were obtained from Waltheri's Go Pattern Search using the full database restrained to a local search (accessed August 2022).
• (Youtube) Michael Redmond: The Direct 3-3 Invasion Keep it Simple! (2020)
• (Youtube) Ryan Li: 10 Outdated Go Joseki and Why (Part 1 of 2) (2020)
• (Youtube) Ji Lili: 纠正我们学过的错误下法：第六讲·二路爬的时机 (CN) (2022)
• (Bilibili) Zhan Ying: AI围棋定式3-最受欢迎的点三三应用秘籍 (CN) (2021)