# What world is this where 6 + 6 = 10?

Upon traveling the world, I found myself in a peculiar place which was called Gauss. The locals there were a peculiar people, as I soon found out. I stopped to grab a bite to eat and learn about the culture. Whereupon I found a man starved for hunger. With sympathy in my heart, I purchased a meal for him and myself, both of which were priced at 6 dollars. I gave the waiter 12 dollars, and immediately, he gave me back 2.

I asked in surprise, "I thought the meals were 6 dollars each."

He replied, "They were, this is your change."

I ate my meal in relative silence, supposing that I was just fortunate. But after the meal, I decided to get gas, priced at 2 dollars. I filled up my tank, which was 14 gallons, and to my displeasure, I owed 30 dollars.

I payed the money unhappily and looked for a hotel to stay. The man there told me it was 90 dollars a night. I decided to book a room for three days, so I gave the man 270 dollars, but he told me the price owed was 300!

As you can imagine, I'm pretty upset. This world of theirs is confusing to me. Can you help me figure out why I keep paying these ridiculous prices? What world exists where 6 + 6 = 10?

HINT:

Just now, I purchased a snack in a vending machine that was 75 cents. It took my three quarters gladly and gave me the toblerone I desired. To be specific it was 25 + 25 + 25 = 75 cents.

HINT 2:

A friend in the village explained some math to me. He said that while 3*25 = 80, 25 + 25 + 25 = 75. Also that 100/15 = 7 and 20 - 14 = 10.

• Kind of reversed world,the rule: if you are serviced by another person, you are given the tip, if you are not serviced, then you give the tip, in case of machine -no tip required ;) – user63710 Nov 21 '19 at 19:43
• @Rob, it's a singular rule, but that rule has different things under it. – Joe-You-Know Nov 22 '19 at 2:09
• @Joe-You-Know Hints should not be used to constrain a puzzle after answers have been added. Please read puzzling.meta.stackexchange.com/questions/6256/… , and then remove the hints which conflict with previously-added answers. – Sneftel Nov 22 '19 at 13:28
• @Sneftel That's not what's going on here. The hint was just to clarify that an answer was wrong and not the intended solution – Quintec Nov 22 '19 at 14:00
• @Quintec Exactly. If your original puzzle had solutions which satisfied the clues but weren't your "intended solution", that doesn't make them incorrect. It just means you accidentally made a puzzle with multiple correct solutions. You should not go back and edit your puzzle to make those solutions incorrect after the fact. – Sneftel Nov 22 '19 at 14:08

Building off of @JS1's answer, I think

calculated numbers (even in intermediate calculations) are rounded to the least number of significant digits in any of the elements of the calculation.

Meals:

2 [1 significant digit] x 6 [1 significant digit] = 12, rounded to 10 [1 significant digit].

12 [2 significant digits] - 10 [1 significant digit] = 2, accurate to 1 significant digit (even though this is not normally how significant digits are used).

Gas:

2 [1 significant digit] x 14 [2 significant digits] = 28, rounded to 30 [1 significant digit]

Hotel:

3 [1 significant digit] x 90 [1 significant digit] = 270, rounded to 300 [1 significant digit]

Snack machine:

25 [2 significant digits] + 25 [2 significant digits] + 25 [2 significant digits] = 75, accurate to 2 significant digits.

The snack machine probably used addition instead of multiplication to calculate the amount of money inserted due to coins being inserted sequentially, rather than handed over all at once.

• This is correct. nice job – Joe-You-Know Nov 22 '19 at 19:59
• more specifically, rot13(Gur fvt svt ehyr vf nccyvrq gb ahzoref 1-9, juvyr gur ahzore 0 8f 9taberq nf n fho svt) – Joe-You-Know Nov 22 '19 at 20:03
• +1 but oof. Who wants to live in a world without distributivity?? A world where: $3(4+20)$ = 60 but $3\cdot 4 + 3\cdot 20 = 70$. – knrumsey Nov 22 '19 at 23:26
• If this is true, then how come 3*25 = 100 and not 80? – MackTuesday Nov 23 '19 at 3:03
• I don't understand why this is correct. According to your rule, the machine should do $25 + 25 = 50$, and then $50 + 25 = 80$. Because (even in intermediate calculations) it should be rounded accordingly. @Joe-You-Know Perhaps you could clarify this? – WhatsUp Nov 24 '19 at 19:14

This town expresses numbers in Base 12.

For the meals:

He pays for two $$6$$ dollar meals, which is $$10(b_{12})$$, or $$12(b_{10})$$. Thinking he is paying the required $$12$$, he pays $$12(b_{12})$$ which is $$14(b_{10})$$, hence the $$2$$ change.

For the gas:

His gas tank is $$14(b_{12})$$ gallons which is $$16({b_{10}})$$ gallons. Since his tank cannot be empty, it has $$15(b_{10})$$ gallons in it. Thinking he has added $$13(b_{10})$$ gallons, but has actually added $$15(b_{10})$$ gallons, he is upset with the gas clerk who accepts $$30(b_{10})$$ of his $$1$$ dollar bills as payment.

For the Hotel:

He booked it for "three days". He arrived Monday evening and stayed Monday, Tuesday, Wednesday, and Thursday nights - three days and four nights. At $$90(b_{12})$$ which is $$108(b_{10})$$, that comes out to $$300(b_{12})$$ for the four nights.

• This is really clever, but not what I had in mind. I honestly can't believe this lined up in this manner. The chances of that happening have to be astronomical. – Joe-You-Know Nov 21 '19 at 17:12
• This answer requires the narrator speak in b12, which is inconsistent with his apparent bewilderment. That said I do think it's what op intended. – MooseBoys Nov 21 '19 at 17:14
• Doesn't work for the vending machine though, as three "25"-cent coins would be worth "73" cents, not "75". – dan04 Nov 22 '19 at 5:37
• @MooseBoys The coins and banknotes are labeled in base 12. When he says he paid \$12, it was a \$10 note and a \\$2 coin. – user253751 Nov 22 '19 at 10:58
• This is exactly what I first thought of too! As the joke goes, "There are 10 types of people in the world: those who understand base-2, and those who don't." – Graham Nov 22 '19 at 15:59

I feel like this has something to do with:

Gaussian rounding, since the town is named Gauss. The numbers are being rounded up or down to one significant digit (like hexomino's answer). In Gaussian rounding, numbers ending in 5 are rounded to the nearest even number in the previous digit. But that doesn't fit with the vending machine, so it may be that in this town, I think one of the following is happening:

1. Numbers ending in 5 are rounded toward the nearest odd number instead.
2. Numbers are rounded toward the nearest odd number no matter what the trailing digits are. For example 19 would round to 10.

There aren't enough examples to distinguish between the two cases above.

The meals:

12 rounds down to 10.

The gas:

28 rounds up to 30.

The hotel:

270 rounds up to 300.

The vending machine:

0.25 rounds to 0.3, and three quarters make 0.9 which covers the cost of the chocolate which was 0.75 (which might be rounded down to 0.7).

• Why would the waiter return 2dollars then? (not 2 rounded down to 0dollars) – JMP Nov 21 '19 at 19:51
• 2 is already one significant digit, no rounding necessary. – Thomas Markov Nov 21 '19 at 20:14
• You were so close, except with the vending machine. Everyone has the general right idea, but at the same time is not getting it. – Joe-You-Know Nov 22 '19 at 2:10
• Integers that represent the count of some item have infinite significance. – DJohnM Nov 24 '19 at 4:03

The residents of Gauss use normalized scientific notation to represent their currency and always round the coefficient on any prices to the nearest integer or the first significant figure (maybe they don't like to deal with non-integers).

Food

The price of each meal is $$6 \times 10^0$$ dollars and two meals becomes $$1.2 \times 10^1$$ dollars which gets rounded to $$1 \times 10^1$$ dollars.

Gas

The price of gas is $$2 \times 10^0$$ dollars per gallon, so $$14$$ gallons would come to $$2.8 \times 10^1$$ dollars which gets rounded to $$3 \times 10^1$$ dollars.

Accommodation

The price of a room for the night is $$9 \times 10^1$$ dollars and so three nights would be $$2.7 \times 10^2$$ dollars which gets rounded to $$3 \times 10^2$$ dollars.

• Interesting, interesting. But, just now, I purchased a snack in a vending machine that was 75 cents. It took my three quarters gladly and gave me the toblerone I desired. – Joe-You-Know Nov 21 '19 at 17:32
• To be specific, I put 25 cents + 25 cents + 25 cents in and got the chocolate I wanted. – Joe-You-Know Nov 21 '19 at 17:36

I think the rule is:

1. If it's an even number, it rounds to the nearest order of magnitude.
- 12 rounds to 10
- 28 rounds to 30
- 270 rounds to 300
2. If it's an odd number unless it ends in 5, it's rounded up to the next even number

Hint one:

Just now, I purchased a snack in a vending machine that was 75 cents. It took my three quarters gladly and gave me the toblerone I desired. To be specific it was 25 + 25 + 25 = 75 cents.

25 and 75 do not round, as they end in 5

Hint two:

A friend in the village explained some math to me. He said that while 3*25 = 100, 25 + 25 + 25 = 75. Also that 100/15 = 7 and 20 - 14 = 10.

- 3 * 25 = (3 rounded to 4) * 25 = 100
- 25 + 25 +25 = 75 (no rounding, ending in 5)
- 100 / 15 = 6.66667 (simply round for decimals?) -> 7
- 20 - 14. 14 rounds to nearest order of magnitude (10). 20 - 10 = 10

A programmer's answer here, and probably not what you want to hear, but one possible answer is:

A base 12 math world! (Where 10_b12 == 12_b10, and numbers are not just 0...9 but eg. 0..9,A,B - like Hex/Base16 but going just up to 11/A.).

• FYI, you are most likely got the downvotes because the most upvoted answer says the same thing. – Lafexlos Nov 22 '19 at 15:48