Math and Logic Puzzles: Redux

[Harambey180]

Here's a maths puzzle from a maths book I just found:

Here you see three squares. Each square has four numbers in the corners; the first two squares have another number in the middle. The last one doesn't. Using the numbers in the corners so that you get the number in the middle in the same way for all three squares, what should the missing number in the third square be?
(Example: If you add up the top left and bottom right corner in the first square, you also add up those numbers for the second and third square)

7		2
	9
6		3

8		1
	7
4		4

9		6
	?
8		5

[Harambey180]

Answer 1: 25. Product of bottom numbers minus sum of top numbers.
Answer 2: 11. Bottom left number + 3.
Answer 3: 5. 23 - 2 * top left number.
Answer 4: 17. 2 * Top right number + 5.
Answer 5: 5. 15 - 2 * bottom right number.
Answer 6: 9. Sum of top numbers, multiplied by π, divided by bottom right number, and rounded down.

Answer 1 is correct and it is the only answer.
I forgot to mention that you have to use all four numbers in the corners and the four numbers only. All your other answers don't use all numbers and some also use illegible numbers.
Great job on finding the answer though! I guess I should have told to only use the four numbers...

[Harambey180]

Here are two more puzzles. They're separate puzzles, but both puzzles work in the same fashion.

For both puzzles: What missing number is supposed to be in the middle of this 'number wheel'? Explain why it is supposed to be the number you got and how you got to that answer.

Puzzle A:

18	10	8
24	?	37
8	22	5

Puzzle B:

13	1	2
2	?	31
17	43	2

[Harambey180]

Puzzle A: Answer is 18 -- Each individual column adds up to 50, so middle square's value = 50 - 10 - 22 = 18.
Puzzle B: By comparison with the previous puzzle, it looks like a significantly more difficult one. I guess if no one answers it, then the fairest thing is to let me give the next puzzle

The answer to puzzle A is not the one I am looking for, even though it is correct in the way I showed it.
You should actually see the puzzles as connections that go through the question mark in the middle. So top left with bottom right, left with right, bottom left with top right, and top with bottom. Not with columns or rows.

[Harambey180]

I would think you would have to give a bit more information to help us out on those puzzles; I don't see any patterns.

Not simple addition, subtraction, multiplication, division (try it and you'll see you get different answers for each)
Not modular arithmetic (i.e.: 18 mod X = 5 does not have the same value X as say if one tried 8 mod X = 8 and 24 mod X = 37)
Not non-decimal base system (Some of the sequences increase, others decrease, and in the first one, there is no change and this stuff implies the middle number would be different for each one)
Nothing to do with prime factorization
Doesn't seem to have anything to do with the digit sequence

Also, how can the same input give two different outputs??? (See puzzle B : Apparently, 2 gives both 17 and 31 as possible outputs after going through the middle)

Oh no no no, it's not a sequence. It's the lines I was talking about, but you end up with the middle number. Not with one of the other two numbers.
So you use the two numbers in such a way each time that the 'third number' is always the same. That's the missing number.

Don't worry, this puzzle has 3 stars out of 3 for difficulty in the book that I got this from, so it belongs to the hardest ones in there. The previous one, for comparison, was 2 stars out of 3.

[Harambey180]

Puzzle A:

18	10	8
24	?	37
8	22	5

Puzzle B:

13	1	2
2	?	31
17	43	2

Puzzle A: the answer is 6.
Add the numbers on either side then take the product of the 2 digits of the result.
So:
18 + 5 = 23; 2 * 3 = 6.
24 + 37 = 61; 6 * 1 = 6
8 + 8 = 16; 6 * 1 = 6
10 + 22 = 32; 3 * 2 = 6.

Puzzle B: the answer is 12.
The product of the 2 numbers then the product of the digits of the result.
13 * 2 = 26; 2 * 6 = 12
2 * 31 = 62; 6 * 2 = 12
17 * 2 = 34; 3 * 4 = 12
43 * 1 = 43; 4 * 3 = 12

Do you own the book

[Harambey180]

Yup, they are! Great job

[Harambey180]

Here's what should be easy ones, as was traditional for me last time:

You've been hired at the state prison and your job is to oversee the factory that makes the state's license plates. A new license plate series started on Monday, January 1, 2018. In the series, license plates have to be three letters followed by three digits. License plates cannot use the letters I or O, nor can the digits be all zeroes. The license plates are made in sequential order, with the very first plate made on the first day being AAA-001, the plate after that being AAA-002, and so on through AAA-999, with the plate following AAA-999 being AAB-001. Assume your prison makes exactly 25,000 plates per day Monday through Friday with no holiday breaks and that there are no other restrictions to the allowed plate numbers.

1. What was the number of the last plate made on January 5, 2018? What will be the number of the last plate made on December 31, 2018?
   
2. On what day will you make plate number ZZZ-999? Which plate that day will that be?*
   
3. Suppose a law is passed that, after all the available numbers as above are exhausted, the alphabetic and numeric portions change places so that after ZZZ-999 comes 001-AAA. The numeric portion is still the portion that is incremented for the next plate, with the rollover from 999 to 001 incrementing the alphabetic portion, so the sequence goes ..., ZZZ-999, 001-AAA, 002-AAA, ..., 999-AAA, 001-AAB, ... If ZZZ-999 and 001-AAA are made the same day and you still make 25,000 plates total that day, when will you make plate number 999-ZZZ?

* For clarity, I'm looking for an ordinal number to the second part of the question, such as "the 12,345th plate" (which is incorrect, by the way).

The end of January 5, 2018, means 25,000 * 5 = 125,000 plates will have been made.
There are 999 plates that start with AAA-???, AAB-???, and so on.
999 * 125 = 124,875 plates, so we are looking for the 126th letter combination, and the 125th number combination of that letter combo, which is 125.
The 126th letter combo is: AA?-??? has 26 combinations. 26 * 4 = 104. We are looking for the 5th A??-??? letter combo, so AE?-???, and we are looking for the 22nd letter to follow that (126 - 104 = 22), which is a W.
That makes the last plate on Friday, January 5th, 2018 to be AEW-125.

December 31, 2018 is the 365th day of the year because 2018 is not a leap year. There are 7 days in a week, and 7 * 52 = 364. So the 53rd week, and December 31 2018 is a Monday.
The plate factory only works 5 days a week. In 2018, there are 52 full weeks, so 52 * 5 = 260 days. Then finally add the Monday on December 31 2018: 261 days.
At the end of December 31, 2018, 25,000 * 261 = 6,525,000 plates will have been made.
There are 999 plates per letter combo. 6,525,000 / 999 = 6531.531..., so we are looking for the 6532nd letter combination.
6531 * 999 = 6,524,469.
6,525,000 - 6,524,469 = 531, so we need the 531st number combination, which is 531.
There are 26 AA? letter combinations, and 26 * 26 = 676 A?? letter combinations.
6532 / 676 = 9.662..., so we are looking for the 10th ?AA letter combination, or the 10th letter as the first letter, which is a J.
-------------------- Right now, we have the plate: J??-531 ---------------------
9 * 676 = 6084 and 6532 - 6084 = 448, so we are looking for the 448th two-letter combo.
448 / 26 = 17.230... so we need the 18th letter as the second letter of our plate, which is an R.
17 * 26 = 442 and 448 - 442 = 6, so the last letter is the 6th letter, which is the F.
That means that the last plate made on Monday December 31, 2018 is the plate JRF-531.

There are 999 number combinations, and 26 * 26 * 26 = 17,576 letter combinations, which means 17,576 * 999 = 17,558,424 combinations.
That means we are looking for the date on which the 17,558,424th plate is made.
25,000 plates are made each day, so 17,558,424 / 25,000 = 702.336, so the plate is made on the 703rd day on which plates are made.
There are five days a week in which plates are made, so 703 / 5 = 140.6, so the plate is made in the 141st work week, and on the third day of that week, so the Wednesday.
7 days a week, so 140 * 7 = 980 days + 3 = 983, so the 983rd day with January 1, 2018 as the first day.
2018 and 2019 are not leap years, so 365 * 2 = 730, and 983 - 730 = 253, so we are looking for the 253rd day in 2020.
2020 though is a leap year. With 31 representing the amount of days in January and so on, 31 + 29 + 31 + 30 + 31 + 30 + 31 + 31 = 244 days January through August. The 9th day of September, so the plate is made on Wednesday, September 9, 2020.
702 * 25,000 = 17,550,000, and 17,558,424 - 17,550,000 = 8,424.
That means that ZZZ-999 is the 8,424th plate made on Wednesday, September 9, 2020.

There are 17,558,424 combinations where the first part is made up of letters. Now we reverse that, so 17,558,424 * 2 = 35,116,848 plates.
35,116,848 / 25,000 plates a day = 1404.67... so the 1405th day is the day the plate is made.
Five days a week in which plates are made, so 1405 / 5 = 281.2, so the plate is made on the first day of the 282nd week.
7 days a week, so 281 * 7 1967 + 1 = 1968, so the 1968th day with January 1, 2018 as the first day.
2018 and 2019 are not leap years, so 365 * 2 = 730. 2020 is a leap year, so 730 + 366 = 1096. And 1405 - 1096 = 309. So the plate is made on the 309th day in 2021.
31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 = 304, and 309 - 304 = 5, so the 5th day of November.
That means that the plate 999-ZZZ will be made on Monday, November 5, 2021.


I really hope I got all of these right because I spent a lot of time on getting these answers.

[Harambey180]

First one is AFF-125 then.

[Harambey180]

Second one is MJD-531 then.

[Harambey180]

Third one is February 12, 2020. It is the 10,176th plate.

[Harambey180]

Last one is March 25, 2022. And it is a Friday.

[Harambey180]

1 3 2 4
2 4 1 3
3 1 4 2
4 2 3 1

All horizontal lines add up to 10.
All vertical lines add up to 10.
Both diagonals add up to 10.
All 2x2 squares add up to 10.

Beat this.

[Harambey180]

1 4 3 2 5
2 5 3 1 4
3 3 3 3 3
4 1 3 5 2
5 2 3 4 1

I'm just too good at these magic squares...

[Harambey180]

4,047	-2,953	2,047
-0,953	1,047	3,047
0,047	5,047	-1,953

I couldn't help doing this (with a generator)

[Harambey180]

Alright, here are two new puzzles!

Puzzle A:
Use the four numbers in the four corners the same way each time to get to the number in the middle. What should that middle number be for the third?

3		2
	30
2		4

6		3
	54
5		1

3		1
	?
8		2

Puzzle B:
This puzzle is the same as the puzzle in post 9. These are two individual puzzles that work the same but have no meaning for the other puzzle.
When you see the puzzle as having four lines, each of which are straight lines going through the middle, how do you use the numbers to get to the number in the middle and what should that number therefore be?

6	1	15
5	?	4
2	16	3

6	3	18
4	?	9
2	12	6

These should be easier than previous puzzles I posted here.