Each cluster of three numbers on the left produces the number in the middle by the same hidden rule. Work out the rule from the known clusters and find the missing output on the right. (Puzzle from the game Rule Hunter)

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Each pair of numbers produces the number below it by the same hidden rule. Work out the rule from the known pairs and find the missing number. (Puzzle from the game Rule Hunter)

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I had an equation come to me in a daydream. I have a very particular reference in mind with it and was wondering if anyone could come to the same conclusion or possibly see something I missed. I'll reveal my thoughts on it after people have some time to see what they find. The equation is 64/15=2+2.

The same rule connects the four numbers in every diamond. Find the rule and the number that replaces the question mark.

You have 2 eggs and a 100-floor building. You need to find the highest floor from which an egg can be dropped without breaking — using the fewest possible drops in the worst case.

What's the minimum number of drops needed?

I made a full video breaking down the solution step by step — link in the comments!

https://youtu.be/LVHoFrc0T-4?si=eEOKQvoKqGNYdjFD

*Clue: Look for sequential patterns.

Each fruit hides a number, the same in every row. Find the missing sum. There is a shortcut that avoids working out each fruit separately.

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Only +, −, × and ÷ are allowed. No concatenation or exponents.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

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I've been presented with a math puzzle that I can't seem to solve.  The puzzle is a variation of the old game of "24" that used to come on cards.  There are six input numbers, and 4 operations allowed.  Each input and each operation may only be used once.  Order of operations applies, so multiplication and division are processed before addition and subtraction.  The six inputs are:

6, 9, 11, 12, 25, and 27.

The 4 allowed operations are:

addition, subtraction, multiplication, division.

The puzzle:  Using each input and each operation no more than once, and following the standard mathematical order of operations, find a way to get the result of 12. At least one operation must be used.

What is the solution? Is there a standard way beside brute force to find the solution?

Thanks

Mathematical Problem: Minimum Number of Vectors in an \(N\times N\) Grid
Consider an \(N\times N\) square grid, where \(N\) is any positive integer.
Rules and Definitions
• Empty boxes: Exactly one box in each row and exactly one box in each column must be empty. Therefore, the total number of empty boxes is \(x=N\).
• Vectors: Every non-empty box belongs to a horizontal or vertical vector. A vector may occupy a single box or multiple consecutive boxes, but it must be straight: it cannot turn or be both horizontal and vertical.
• Counting vectors: A vector is a maximal uninterrupted straight run of non-empty boxes. Consecutive non-empty boxes in the same row or column may belong to one vector; separate runs count as separate vectors.
• Objective: The positions of the empty boxes may vary, provided the rules above are satisfied. Vectors may be horizontal, vertical, or a mixture of both.
The Question
Let \(y\) denote the total number of vectors and \(x=N\) the total number of empty boxes.
Is it true that (y>x) for every valid configuration whenever ( N > 3 ), regardless of the grid size or the arrangement of the empty boxes?

The same rule turns every pair into the number below it. What is the missing number, and what is the rule?