# Digital Bananas

Homer is taking a joy ride on the highway when he is pulled over by officer Mickey. "Do you know how fast you were going?" the officer asks condescendingly. Homer replies uneasily, "I may have been going a little fast, maybe 74 miles per hour...". The officer sighs, "101 miles per hour, actually. You are aware that the speed limit is 70, yes?" Homer nods. "Well I'll let you off with a warning this time," continues the officer, putting away his notebook. "Next time, I'm going to have to ticket you."

Why did the officer let Homer off with only a warning?

Small hint:

The tag would be appropriate for this question.

Big hint:

"When this baby hits 130 miles per hour, you're gonna see some serious sh%&..."

• vulgar, as in hint 3, is usually prohibited. please consider deleting it since this is supposed to be a safe-for-children site. thank you for your cooperation – Omega Krypton Sep 26 '19 at 8:05
• @Omega The quote is from a PG-rated film, so it seemed fair game. – MooseBoys Sep 26 '19 at 16:45
• Aha, that seems much more appropriate. Thanks for your cooperation! – Omega Krypton Sep 26 '19 at 23:44

Homer was running only $$9_{(10)}$$ mph above the speed limit so officer Mickey let him go.
The speed limit was $$70_{(8)}$$ (or $$56_{(10)}$$) while Homer was going $$101_{(8)}$$ (or $$65_{(10)}$$).

The explanation for the title

and the base of their numbering system is that in Mickey and Homer's world, everyone has 8 banana-looking digits on their two hands.

• On the right track. The title explanation is close but not quite correct. The missing piece is related to why you shouldn't need the spoilers. – MooseBoys Sep 26 '19 at 15:03
• @MooseBoys So, this video doesn't explain the 8 vs the 10? It seems like that is what greysaff was implying. Edit: Oh I just read the comment on the other answer. I guess the "bananas" part is what is missing. – JS1 Sep 26 '19 at 20:19
• A clue for the 8 fingers is that both Mickey Mouse and Homer Simpson have 4 fingers in each hand – Luis Rico Sep 27 '19 at 7:39

Given the hint,

which is an obvious reference to the first Back to the future movie, except that 88 has been replaced with 130, we can deduce that the implied base is not 10 for all numbers. Indeed, $$\color{red}{88_{(10)}}=8\times 10+8\times 1=\color{red}{130_{(8)}}=1\times 8^2+3\times 8+0\times 1$$

In general, a base $$8$$ number looks smaller when turned into a base $$10$$ number. In particular, $$101_{(8)}=65_{(10)}$$ which is less than the $$70$$ limit.

However,

This is only a partial answer since it doesn't explain how to get the number $$8$$ without the hint (hints are not supposed to be necessary to solve a puzzle), and it doesn't imply why the officer would need to fine the driver next time, since $$65$$ would still be less than $$70$$.

• On the right track. The hints are not necessary, though I did consider adding the 'knowledge' tag at one point. – MooseBoys Sep 26 '19 at 7:04
• The officer says $70_8=56$, so Homer is still speeding. – JMP Sep 26 '19 at 7:04
• Re hints being necessary: This answer was my first idea without even seeing the hint, just from the fact that $101$ is apparently close to $70$. – JiK Sep 26 '19 at 10:50
• @Arnaud I think you're misinterpreting my comment on Greysaff's answer. The fact that Mickey and Homer have 8 fingers, combined with the otherwise unusual exchange between the two parties, is what leads to the conclusion to use base 8. The reasoning for the title is also related to knowledge of the named characters. – MooseBoys Sep 26 '19 at 16:57
• A quote from Walt Disney is rot13(Hfvat svir svatref jbhyq unir znqr Zvpxrl’f unaqf ybbx yvxr n ohapu bs onananf) – Bananenkopp Sep 26 '19 at 20:06

I arrived at the same answer as greysaff after the two hints; here's my take on the title meaning:

A banana bunch is often referred to as a "hand", while individual bananas are referred to as "fingers". Fingers are also "digits" so the title itself is a hint to consider that the two characters only have 8 fingers.

• Digits has two connections to the problem, one of which you've identified. Your bananas connection is not the one I was thinking of. – MooseBoys Sep 26 '19 at 19:35