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76

Oo, this is a good one. Let's do some analysis: We can see the centres of three sides, and the relative positions of the centres cannot be changed, so we know that when/if this cube is solved, the blue side will be adjacent to green and orange. We can also see a blue-white edge piece, and a blue-yellow edge piece as well. This means that as long as the cube ...

44

Most Sudoku puzzles published have only one solution. If there is more than one solution, it is probably a mistake. That said, puzzles with incomplete clues can have multiple solutions. In the extreme case, a puzzle with no clues has 6,670,903,752,021,072,936,960 solutions according to Wikipedia. I don't know if it's possible to have exactly 3 solutions, ...

32

Well so you could do Full solution (The "Try it online" link agrees that this is legal)

28

I have seen two dissenting opinions on this subject (and in my opinion, the first option is right): By definition, a Sudoku has only one solution. Anything else is just a grid of numbers. Sometimes, there are errors in a publication, and a starting grid has multiple solutions, but, then the starting grid was not a Sudoku! From Wikipedia: The number of ...

27

I'm not sure if you don't understand the example puzzle, or if you don't understand how to find the solution. I'll explain both, just in case. The example takes a word/phrase that is 13 letters long with no repeated letters. That means it contains exactly half of the letters in the alphabet (26 total). Then the 13 remaining letters are written below, in ...

25

We can see Therefore But the centre pieces tell us that Which is a contradiction. Therefore this cube cannot be solved.

24

At least not if the solution space is a countable infinite set, or smaller. In that case, we can compute the finite time person 2) will need to find the solution found by person 1) by enumeration, making the strategy of 2) a computable function and therefore violating the constraints of the question. Response to edit: The "does not have to be perfect-...

21

This puzzle is called a "Baguenaudier" which is French for "Time Waster". This solution assumes that you are holding the handle end on the left. The rings are numbered from right to left starting at 1. The solution to the puzzle involves a couple possible moves. 1) The first ring is always available to put on or take off of the bar. You take it off the ...

21

I can sympathize. This is from The Guardian, 2015: From left to right, starting with image 2, we have: no outline not a square green small leaving image 1 because it is the only image that is not an odd one out! Which I think qualifies it as an odd one out. So the answer given might not be unique, which makes some of the puzzles unsolvable.

20

A random configuration of gears is solvable 1/3 of the time* For details on what kind of configuration is solvable, refer to the claim after the image below. Your hunch is right, we can use modular arithmetic here. Let's mark each gear with a letter, starting with the top one as A, going from top to bottom, left to right, as in this figure: Let's denote ...

20

In the classic book "Winning Ways (for your mathematical plays)" by Berlekamp, Conway and Guy there is a small section devoted to wire puzzles (like the Tavern puzzles) in the second-to-last chapter. They use a technique they call a "magic mirror". Imagine a fun-house mirror that distorts your view of the puzzle. Some bits of the body of the puzzle get ...

19

There are several types of clues embedded in these puzzles. Cryptic clues where the words have no meaning. One is meant to break words apart into constituent letters and then reconstruct. The clues are literal, however they are veiled in metaphor and use of hyperbole and exploitation of synonyms. The clues are totally descriptive (easiest, most obvious and ...

18

This is one of the most well-known wooden interlocking "Burr" puzzles, called The Chinese Cross. If you look at the two images below, you can see pieces that are the same shapes as yours, but with each one a different color. Using the colors as a guide, you can see how they fit together in the assembled version. The photos are from Rob Stegmann's page on ...

16

The angels should play the game, as long as the formula for the computable function must be fixed in advance and finite. Proof: let F be the shortest encoding of the formula (length l(F)) using a set of symbols S of size n(S). Then the guessing angel can simply count up from 0 in base n(S), and some time before n(S)l(F) they will encounter the formula. (...

16

This can be solved in 16 steps. Lets use binary here, 0 is off, 1 is on the handle You start with 11111 1 - Remove the first ring to get 01111 2 - Remove the third ring to get 01011 3 - Put back the first ring to get 11011 4 - Remove the first two rings to get 00011 5 - Remove the fifth ring to get 00010 6 - Put back the first two rings to get 11010 7 - ...

15

The easiest explanation would be that in a 3x3 cube, only one cube is out of position, but in a 4x4 cube two cubes are out of position. In a 15 puzzle (the sliding puzzle where you try to put the numbers in order) half of all possible initial positions are unsolvable. They call the solvable positions "even" and the unsolvable positions "odd". The "odd" ...

14

2x2 - area 108 - optimal 2x3 - area 72 - optimal 2x4 - area 108 - optimal 3x4 - area 84 - optimal 1x5 - area 54 - optimal 2x5 - area 304 - optimal 3x5 - area 576 - optimal 2x6 - area 240 - optimal 1x7 - area 1034 1x8 - area 432 - optimal 3x8 - area 5880 1x9 - area 585 - optimal Note that by subdividing the yellow rectangles: 2x3 indirectly solves ...

13

Some simple strategies I've found are: Look for zeroes. Any time you have $X + Y = X$ in the ones column, you know that $Y = 0$. Look for doubles. Any time you have numbers of the format $A + A = B$ in the ones column, you know that $B$ is even, and you know that either $0 < A < 5$ OR that there is a carrying one to the next column. Look for ...

12

By definition, all valid Sudoku puzzles should have only one solution. In point of fact, many of the techniques used for solving puzzles depend on there being only one solution. All of the Unique Rectangle techniques for example, only work if there is one, and only one, solution.

12

Does an algorithm exist? Yes. Consider every valid state of the Rubik's cube. It can be brought to the solved state in 20 moves or less. For each state, apply the sequence of moves followed by its inverse. This giant algorithm is guaranteed to solve any cube. Now, does a reasonable-length algorithm exist? No. I will show that any such algorithm should ...

12

The coding phrase is: The resulting codex is: Found by:

12

The Situation that you are talking about is a lot more common than you know. And it's not only limited to odd-one-out puzzles. There are riddles which are basically "guess what I am thinking" puzzles. Odd-one-out puzzles usually land in this basket. There is nothing wrong with your thinking or your reasoning. The fault, almost always, lies with the question....

12

Probably not. The bin-packing problem is NP-complete, and any instance of the bin-packing problem can be turned into a 2D packing problem. (Represent each integer $k$ with a $1\times k$ rectangle, and make the bins of size $n$ into several $1\times n$ holes with tiny "channels" connecting them into one full shape.) Therefore the 2D packing problem ...

11

This puzzle is indeed broken, and has no solutions. (I ran into the same contradiction as Arnaud.) The organizer, Serkan Yürekli, made a post on the Logic Masters Deutschland forum around that time, confirming this. The 1,1,3 clue at the top was the problem, and because of this they ignored the top arrow when checking solution keys. (In addition, the ...

11

No. If there is exactly one way to fill out the grid in a valid way, then that can be found without using any logic based on uniqueness. (It may be very painful, and require brute-forcing a lot, but it won't be impossible to find.) If there are exactly two valid ways to fill out the grid, then uniqueness logic cannot help you. You can never make a deduction ...

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