The concept of 「ボケ」and 「ツッコミ」is a common concept in Japanese comedy, and one you may be familiar with if you’ve watched British double acts or Japanese manzai (漫才). It’s a classic setup where one person plays the funny man (ボケ, boke) that does / says something stupid, and the other plays the straight man (ツッコミ, tsukkomi) and acts as the voice of reason.
Throughout the series, Ishigami has almost always been the straight man (ツッコミ) to the rest of the StuCo’s antics, but this time, he’s switched to the other role. As such, the theme of this chapter has the StuCo trying to get him back.
Page 6, a
A portmanteau of sumo wrestler「お相撲さん」(osuMŌSAN) and oxygen「酸素」(SANso). Not a particularly high-level pun, Ishigami just takes the “san” from o-sumō-san and combines with the san from sanso to form「お相撲さんそ」(osuMŌSANso).
Page 6, b
This is a reference to the Japanese comedy duo, COOLPOKO (クールポコ). More specifically, it refers to this one line from a particularly famous act, which has since been performed on several platforms since its first performance over a decade ago.
In the original, the lead-up would’ve been something along the lines of:
Fujiwara: “None of your jokes have been landing this whole time!”
Ishigami: “Eh, really...?”
“They haven’t been landing...?”
“I’VE REALLY DONE IT NOW!!”
Due to how obscure this meme is, a strikingly similar substitute was found. All that was left to adjust the setup text to make the joke land. If you didn’t get the joke at first, that’s fine; I doubt that the average Japanese teenage reader would either.
Page 6, c
Again, Ishigami is quoting an old Japanese comedy duo, Shupei and Taiyu Shoinji. In their performances, they’d often use this line, 「時を戻そう...」(Let’s go back in time...) as a bridge between their skits. As for why Ishigami knows so many old references, I have no idea.
As for the English parallel reference, there was only one clear option.
Shirogane’s pun here is just a pair of homophonic noun / verb pairs, i.e., something roughly as funny as “My bread got bred... and my board got bored!” The literal translation of what he said would be:
「ふとんがふっ飛んだ...！」 My futon was blown away...!
「ストーブがすっ飛ぶ！」 And my space heater flew away!
The space heater here is more of a stove-like appliance common in Japanese homes, since central heating / cooling is very rare.
This was substituted for an equally awful joke.
Here Miko prompts Fujiwara for a random word, to which she gives “student council”. The riddle then goes as follows:
“What does the student council and Tsubame-senpai have in common?”
“They both have ninki!”
Here, ninki refers to the pronunciation of both:
「人気」(popularity, for Tsubame-senpai), and
「任期」(term limits, for the student council).
It’s a very clever riddle, and unfortunately, one that is quite hard to translate into English, as we don’t have quite as many homophones as Japanese.
Shirogane's use of「解なし」(kainashi, "no correct answer") here is likely a pun on 「甲斐なし」(kainashi, "efforts were in vain"), which is what Isonokami-no-Marotari said upon finding that he had obtained not the swallow's cowry but rather a handful of droppings in the original Tale of The Bamboo Cutter. This, sadly, had to be lost in translation.
Bonus: joke variants from translation drafts
Q: What does the student council room and Tsubame-senpai have in common?
A: They both have hair, except the student council room.
Q: What do the student council members and Tsubame-senpai have in common?
A: They all see Ishigami as a friend!
Q: What’s the difference between Tsubame-senpai and the student council?
A: One’s pretty, the other’s pretty immoral.
Q: What does the local farmers' market and Tsubame-senpai have in common?
A: They’ve both got huge melons.
Q: What’s the difference between good grades and Tsubame-senpai?
A: Ishigami can get good grades if he tries hard enough.