You and a fellow prisoner are imprisoned in a dungeon and are facing execution. As is usual in these problems, the prison warden has both a love of recreational mathematics and an unusual amount of judicial latitude when it comes to deciding your fate. They want to give you and your cellmate a chance of freedom but don’t want to make it too easy for you. Learning should be fun and on Wordgames.com you can find lots of games like Free The Key that make practice a joy. Play now one our our best puzzle games! So all we need to do is the following. Scan the board from top to bottom calculating a cumulative value which starts as the vector 0000. Each time we encounter a dark coin, we perform an exclusive OR of our square’s location label with the current cumulative value to get the new cumulative value. At the end, this cumulative value represents the location of a square. If we use the board in Diagram 9, this turns out to be 1011. However we need the board to represent the third column of the second row or 0110. To list the bits that need to be corrected, we simply exclusive OR these two vectors together to get 1101 which is the location of the coin that we need to flip. Diagram 1: A Chess Board where a key is placed in a secret compartment in one square. For the purposes of this article, we’ll be assuming the 3rd column of the 2nd row. Introduction Now we’re ready to scale up to bigger boards and consider various approaches. These approaches can be read separately if you want to dip in and out of this article. Approach A: Hypercube

In some sense, the chessboard must encode the 4 bits of information required to specify the row and column of the key square. Given that the prison warden will have chosen a configuration either at random or maliciously, some or all the bits encoded may be wrong. So it must be possible to choose a coin to flip (represented here by the counter changing color) that will fix the wrong bits without impacting the correct ones. In the case where the chessboard already encodes 4 correct bits, there must be at least one square that has no effect on the 4 bits when its coin is flipped. On the chessboard in Diagram 13, we use combinations of letters to label the rows and columns. For example A by itself represents the 2nd column; B by itself represents the the third column; A and B together represent the 4th columns and no A or B indicates the first column. This means that A can indicate either the second or the forth column. The squares are also labeled with combinations of letters. By analogy with the leap year rule we have the following (involving all the squares that contain A in their label): Play using the left mouse button if you are using a computer, or tap the screen if you are using a smartphone or tablet. For some games a keyboard is required. You can also find in-game instructions.The prisoner warden gets to set all the coins on the board and we only get to flip one to convey our message- a tiny signal in all that noise. At first glance this seems completely impossible but let’s start by understanding what is relevant in our puzzle and what we can discard. Then we’ll see if we can start with a simpler version of the problem.

They have a chessboard where each square is covered by a coin — either heads or tails. Moreover it’s a special chess board with a hidden compartment in each square. A single one of these squares contains a symbolic key to the jail and freedom for you and your cell companion. You will know which square contains the key and your fellow prisoner has to guess.We can change from odd to even for our dimension or visa versa by flipping either of the 2 coins. Even nicer, the dimensions are of course independent. There is a coin for every single combination of dimension coordinates. So we can work out the coordinate of the coin to flip (left or right) separately for each dimension. If the current parity coin for a dimension Dₓ does not indicate the location of the key in that dimension, we can change the parity by flipping a coin that is on the right for that dimension: Dₓ =R. If the parity already indicates the location of the key in that dimension, then we flip a coin on the left: Dₓ =L. When we have done this for all available dimensions, we will have unique coordinates for the coin to flip. So to sum up, the chessboard is the set of all possible subsets of the set of 4 elements- the power set of {0,1,2,3}. For a set of n elements, the power set contains 2ⁿ elements and so there are 2⁴=16 squares on the chessboard , one for each subset. Moreover the dark coins select a set of these subsets which is in fact a subset of the power set- one of 2 To ease into this approach, keep in mind the rule for leap years which has a basic condition modified by subsequent conditions: The rules for B columns and C and D rows are very similar. English is a little vague when it comes to logic but X unless Y could be regarded as X and not Y. So if X and Y are invited to a party but really don’t like each other we might have the rule X unless Y and Y unless X, more formally (X and not Y) and (Y and not X) which is exclusive OR. Because our Xs and Ys are disjoint, the full rule of X will give us one half of the exclusive OR and the full rule for Y will give us the other half. Let’s imagine our squares to be arranged into a 2x2x2x2 tesseract — a 4 dimensional hypercube of side length 2.

One way of looking at exclusive OR is as a controlled NOT with a control argument and a data argument. When the control argument is zero, nothing happens to the data argument. However when the control argument is one, the data argument gets inverted either from 0 to 1 or 1 to 0. Finally the exclusive OR operator available in many computer languages doesn’t just operate on a single bit. It can operate on two bit vectors where the corresponding individual bits in each vector are exclusive ORed to get the result. As with all vector spaces, a general vector here is a linear combination of the independent basis vectors. However the scaling factors are all either 0 or 1 which implies that each basis vector is part of the linear combination or it is not. In effect, this vector space describes the 2x2x2x2 tesseract from Approach A. Resident Evil 4 Remake: Church Key Puzzle - How To Get Apostate's Head And Blasphemer's Head Apostate's Head

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You and your cellmate can discuss how to encode a message using the chessboard but the prison warden can hear and understand everything that you say. Your companion re-enters the room, without having any opportunity to see or communicate with you. He observes the chessboard and the arrangement of coins and points to the square where he believes the key and freedom reside. Diagram 2: The chessboard from Diagram 1, now covered in counters after the secret compartment has been closed Some Properties of the Problem You observe the prison warden hiding the key in one square and then arranging the 64 coins as heads or tails however they deem fit — presumably trying to frustrate your system.