Deluxe lights out game

Designed for solo bridge players looking to improve their game. Comes with Saitek score pad. Manuel is missing but available online. Utilizes four AAA-battery. LED Display. Label on rear is faded. Utilizes four AA-batteries. One or two player game modes. Black LCD display. Looks great with only minor wear to the back and works perfectly. Utilizes four AA-batteries, and has no corrosion in the battery compartment and compartment lid intact. LCD hand held video game with Talk. Some yellowing on housing.

Utilizes two AA-batteries, has no corrosion in the battery compartment and compartment lid intact. Screen may look scratched, but only due to photos. Challenging memory game! It has lighted buttons. Looks great with only minor, normal scuffing.

Utilizes three AA batteries, has no corrosion and still has battery door intact. Made in China. Super challenging memory game! This game unfortunately has an operational issue. Utilizes three C batteries, has no corrosion and still has battery door intact. Pong like game for one or two players. Looks great with only minor scratches on face and works perfectly.

Utilizes two AA-batteries. Electro-mechanical game. Two speed single player game with two levels of difficulty. Utilizes two C-batteries. Battery compartment clean and perfect, no corrosion. Electronic Video game. Single player game with two levels of difficulty.

The face has notable wear but the unit works perfectly. Utilizes four AA's. Battery compartment has some corrosion. LED display. The contacts strips themselves are no longer available in most cases, but if your rubber matrix is still intact you can easily do a quality repair! Probably could have done many.

Once mixed, compound must be used with-in 72 hours. Great for musicians, repair shops and individuals needing to resurrect those old remote controls. We have tested this stuff in our own repair shop and it works great. Manufacturer has tested repaired contacts to over , keystrokes and they passed. If you've tried cleaning yours, they work for a while but they just keep getting worse, here's your fix.

We were amazed at how well this stuff works. Comes with instructions and everything you need. Very un-impressive looking package, but makes up for it in performance! You should never open any electronic unit without proper knowledge of electronics and the hazards of electricity! Battery compartment missing lid however no corrosion. Very Rare. There are 5 playing levels.

Which level will be played depends on which corner is pointing up when the puzzle is switched on, and level 5 is only accessible when level 4 has been solved. When playing the game the corners will show various colors, and they change depending on the level when you rotate the puzzle letting a different corner point upwards. The aim is to make all the corners red. No cracks or chips to the black body or any of the plexi windows. All LED's function.

Utilizes four AA, 1. Battery door intact and latches. Evidence of past, but cleaned corrosion on door terminals. Very minor and no affect on operation. Made in Taiwan. For one or two players. Utilizes two 9V-batteries. Label on back with instructions is in excellent condition. Eight different games. Red LED screen. Utilizes six AA volt battery, has no corrosion and still has the battery door.

Abbreviated instructions on back. Made in Beverly, MA. Battery compartment in perfect condition, no corrosion. Heat Seeking Tongue action. Utilizes 2 AA-batteries. Original version with sound. Read more about the condition Used: An item that has been used previously. See all condition definitions opens in a new window or tab.

Does not apply. Shipping and handling. The seller has not specified a shipping method to Germany. Contact the seller - opens in a new window or tab and request shipping to your location.

Shipping cost cannot be calculated. Please enter a valid ZIP Code. Shipping to: United States. No additional import charges at delivery! This item will be shipped through the Global Shipping Program and includes international tracking.

Learn more - opens in a new window or tab. There are 1 items available. Please enter a number less than or equal to 1. Select a valid country. Please enter 5 or 9 numbers for the ZIP Code. Handling time. Will ship within 5 business days of receiving cleared payment.

The seller has specified an extended handling time for this item. Taxes may be applicable at checkout. Learn more. Return policy. Return policy details. Seller does not accept returns.

Refer to eBay Return policy for more details. You are covered by the eBay Money Back Guarantee if you receive an item that is not as described in the listing. Payment details. This leaves only tilings which have no monominoes and no vertical dominoes, i. If n is even, then there is exactly one such tiling left.

Like before, any tiling that is not symmetric about this line can - together with its mirror image - be ignored, since we only need the parity of the number of tilings. We only need symmetric tilings.

As the line of symmetry goes through a row of squares, there can be no vertical dominoes on this row that would be asymmetric. For this to be odd both factors must be odd, so from the previous examples we see that n must be even and also not 2 modulo 3. The number of tilings is therefore odd exactly when n is 0 or 4 modulo 6. Proof: Click to view proof.

Proof: Click to hide proof. This gives the patterns below left. These are not easily analysed in general, and sometimes give rise to more complicated quiet patterns. Lets do some simple cases. In fact we can do the trick twice, along the two diagonals of the square. For a tiling to be symmetric about the diagonals, there can only be monominoes on those diagonals.

These must all be the same due to symmetry, and they can be filled with one domino or two monominoes. There are then exactly two symmetric tilings, which means that R 4,4 is even.

Therefore there are quiet patterns, such as the ones below. These pairs can be ignored so we are left with patterns which have no vertical domino nor two vertically adjacent monominoes in the second and third rows. The bottom left corner has a domino in it, either vertical or horizontal. If it horizontal with a horizontal domino above it then there are S n-2 ways to complete the tiling.

The final possibility is a horizontal domino with a monomino above it. Now the next monomino in the top row can be immediately adjacent to the first giving S n-2 completions or there may be one horizontal domino first giving S n-4 completions or two S n-6 , three S n-8 etc. By repeatedly plugging this into the equation it becomes clear that S n is even exactly when n is 4 modulo 5 i. The same techniques don't work well for wider rectangles as it gets messy very quickly.

What can be said is that for every fixed m, as n increases the parity of R m,n will show a repeating pattern. The following two sections have quite a lot of theory behind them. It would take too much space to go into full detail, so I will skip some proofs and explanations. These sections work up to one of the most important theorems about Lights Out on rectangular boards.

Its determinant, calculated in the same way as before, is then some polynomial in x. Each row of the matrix has only one item with x in it, so each term in the determinant has degree of at most n, and only the diagonal term in fact does have degree n.

The determinant det xI-A therefore is a polynomial of degree n. It is called the characteristic polynomial. This matrix therefore is not invertible, and has its own 'quiet pattern', i.

We call k an eigenvalue , and v a right eigenvector or column eigenvector. There are n roots to the characteristic polynomial, though some may be repeated. For each root, i. If the eigenvalue is a repeated root of the polynomial, then there will be just as many independent eigenvectors associated with it as the number of times that root is repeated. This means that there are in fact exactly n independent column eigenvectors v i , and similarly exactly n independent row eigenvectors w i.

An interesting fact about the characteristic polynomial c x of matrix A is that A itself satisfies the polynomial, i. This can only mean that c A is the zero matrix. In other words it has ones only at entries adjacent to the diagonal. This type of matrix will be useful later on. The characteristic polynomial c n x for matrix A n is fairly easy to find as there are so many zeroes. You can see this in this example for A 5 :. The polynomials in this sequence are similar to Fibonacci polynomials.

One reason that these are named after Fibonacci is that f n 1 is the Fibonacci sequence 0,1,1,2,3,5,8,13, If we are not working modulo 2, then we do have to subtract. Those polynomials are called normalised Chebyshev polynomials of the second kind. In the mathematical literature you will see both names used.

These matrices B n are the same as A n except that they have ones along the diagonal. Its characteristic polynomial is now easy to deduce:. We can now put these special matrices to use. Consider a standard Lights Out game on a rectangular board.

Except for light chasing, in all the previous sections we never used the specific shape of the board to simplify matters. Light patterns or button patterns on a board were then long vectors without any board shape context. This time we use rectangular matrices of the same shape and size as the board. Such a matrix can represent either a light pattern or a button pattern.

We would like to have some expression for the effect of this pattern on the lights, and the special matrices A m and B n will be used for that. Consider XB n. It is as if XB n is the result of the button presses if only vertical neighbours and the button itself were affected by each button press. Similarly, now consider A m X. Thus A m X is the result of the button presses is only the horizontal neighbours and not the button itself were affected by each button press. This kind of matrix equation is known as Sylvester's equation.

Of course the question arises of how to solve X , but I will not discuss that here as the previous methods work well enough for rectangular boards, especially if light chasing is done to reduce the size of the matrices. A more interesting question that can be answered from this equation is, when does X have a unique solution? Therefore if there are no quiet patterns other than the trivial pattern no buttons pressed , then all solutions are unique. To answer the question of when it has non-trivial solutions, we again need a bit more general matrix theory.

Let k be an eigenvalue of matrix A , and lets suppose that -k is an eigenvalue of B. Then we can find a column eigenvector v of A with that eigenvalue k, and a row eigenvector w of B with the eigenvalue of -k. We now find that:. This started with the assumption that -k is an eigenvalue of B for some k which is already an eigenvalue of A. Note that the sharing of a common root k means that the polynomials char[A] x and char[B] -x have a common factor x-k. One thing I glossed over is that in our usual setting the numbers we are working with are all modulo 2.

While a polynomial of degree n has n roots in the complex numbers, with the numbers we have now there might not be any roots. If the two polynomials char[A] x and char[B] -x share the linear factor x or x-1 then obviously the construction of Y above works, using the eigenvalue of 0 or 1 respectively.

In such a case they still share a root even though this root lies outside the numbers we are working with. Even then there will be a non-trivial Y that solves the matrix equation. The converse is also true, but I will not prove any of this here. If we put all this together it does lead us to the following theorem:.

In fact, the number of linearly independent quiet patterns is equal to the degree of the largest common factor. Sketch of Proof: Click to view proof. Sketch of Proof: Click to hide proof. Suppose c m x and c n -x-1 have a common factor of degree d.

In other words, char[A m ] x and char[B n ] -x have a common factor of degree d. This common factor has d roots, which are eigenvalues of matrix A m , and their negations are eigenvalues of matrix B n.

Corresponding to these eigenvalues we have d linearly independent column eigenvectors for A, and d row eigenvectors for B. Proof of converse omitted. As stated above it applies to all rectangular Lights Out versions, regardless of the number of states that each light can have.

Here are the first few of these polynomials, factorised. I have not reduced them modulo 2, but if you were to do so they would often factorise further. This means that we can construct a quiet pattern simply by multiplying the row and column eigenvectors with eigenvalue 0 as explained before. If we instead use the common factor of x-1 and multiply the eigenvectors with eigenvalue 1, then we get the same pattern but rotated.

Note that we did not reduce modulo 2, so this same pattern works with any number of light states, such as the Lights Out with 3 states. There must therefore be a quiet pattern, again applicable regardless of the number of states of the lights. The shared factor is of degree 2 so we cannot simply multiply out the eigenvectors, but the quiet pattern is easily found by a little trial and error. I have also shown a decomposition of that matrix into a product of two rectangular matrices. The polynomial factor had degree 2, so the rectangular matrices the pattern decomposes into have width resp.

It is difficult to explain clearly why this is so. The eigenvalues used here involve square roots since they are roots of a quadratic , and so the eigenvectors would also involve square roots. By taking linear combinations of those vectors we can get two other vectors that span the same space but without the square roots. Those two vectors are used in the columns or rows of the matrices.

The particular quiet pattern derived from the eigenvectors is:. As remarked on before, it is quite easy to generate quiet patterns for larger rectangles by sticking together several copies of a quiet pattern on a smaller rectangle.

This way of repeating a quiet pattern with a strip between them one cell wide to get a quiet pattern for a larger rectangle works very generally.

In the above example the pattern was symmetric. It works just as well for asymmetric patterns provided any two adjacent repeats are mirror images of each other. This ensures that the strip between them is affected equally by the buttons from either side and hence shows no lights as required. At least one sixth of all grid sizes therefore have quiet patterns. These are marked red or orange in the table above. Here are some of the quiet patterns for these boards:. I have now recalculated the table above using the polynomial method up to by A by part of it is shown in the picture below.

Each black square represents zero quiet patterns, i. The shade of the coloured pixels range from red to blue depending on how many quiet patterns there are relative to the shortest side of the rectangle. In [GKW] it is proven that the square Lights Out board with edge length 2 n has a quiet pattern, as well as the square of size 2 n You can see in the picture that these sizes do indeed have more than the average number of patterns.

Suppose you have a board that is n squares wide but indefinitely long with nothing lit, and you press some buttons in the top row. You can then chase the lights down. The button pattern you press on some row matches what the lights were on the row above, which in turn is determined only by the button patterns on that row and the previous row.

As there are finitely many button patterns for two rows 2 2n in fact , you will repeat yourself eventually, after no more than 2 2n steps. By working back upwards you can show that the repetitive pattern begins at the top, i. If you imagine a row zero above the top of the board, in which you pressed no buttons, this is the same as row k because of the repeated pattern.

This proof can be found in more detail in [GTK1]. Click on the link above to see the huge by version. From that picture it becomes very clear that something interesting happens around coordinates that are one less than powers of 2. There seem to be diagonal crosses centred at such coordinates. In August David Beckman sent me a proof for the existence of some of these crosses, using properties of Fibonacci polynomials modulo 2.

From Theorem 8, we know that Q a,b is equal to the degree of the largest common factor of c m x and c n -x-1 , i. Using this we find:. This theorem shows why the diagonal crosses appear on the main diagonal in the image above. I have no doubt that a slight variation of this will also explain the other diagonal crosses in the picture. This is a similar table for the Lights Out game on a torus, i.

These cases have been coloured yellow or orange in the table above. The cases derived from the former are coloured red or orange in the table above, the latter is coloured blue in the table. The theorem below was proved algebraically in [ST] , but here I give an easier to understand proof. Note that this does not depend on the shape of the move. It works with any move shape, as long as it is the same shape for every button on the board. In [ST] , another interesting condition about the dimensions of torus boards with quiet patterns is proved, and I will repeat it here without proof.

The reverse of this result is not true, so the condition on the multiplicative orders is necessary, but not sufficient for there to be a quiet pattern. The rectangular patterns I have found so far are easily verified to satisfy the theorem as shown in the following table.

This is a similar table for the XL knight's game. Note that in general the quiet patterns for this game cannot be tiled as a pattern usually affects two layers of squares around it, and these are effects are not automatically cancelled by a mirror image of the pattern on the other side of those layers. The maths of the Lights Out game is just the same, except that it works modulo 3 instead of modulo 2. Pressing a button now increases the value of the adjacent lights, where a green light has value 1 and a red light value 2 or if you prefer The solution method is the same.

Unfortunately theorem 1 no longer holds, so there are reflexive symmetric games that are unsolvable, i. The solvability test still holds except that it is a bit more complicated: Start with a running total of 0. Consider a quiet button pattern. If a button is pressed once, then add the value of that light to your running total. If a button is pressed twice, then add the value of that light to your running total twice. Unpressed buttons are ignored.

Do this for all buttons. If that total is not a multiple of 3, then the pattern is unsolvable. If the light pattern passes this test for all quiet patterns, then it is solvable. The table below shows the results.

These have a quiet pattern with a number of button presses not a multiple of 3, so that the solvability check on the position with all lights on will fail. The quiet pattern looks like this, and can be extended indefinitely horizontally and vertically:. The bad quiet patterns are based on the patterns below. They can be extended like a checkerboard in both directions just like the previous case.

There are some button patterns for which the affected lights are exactly the same as the buttons pressed. Such patterns are eigenvectors of the matrix A. It is interesting to note that these patterns can be thought of as the quiet patterns of a Lights Out game with matrix A-I , which is a non-reflexive game where each button changes its neighbours but does not change its own light. The eigenvectors are actually found more easily by just playing the game.

Press any button on the top row, and chase the lights down in such a way that the each row has just those lights on in the buttons that were pressed. You will find that the last row also has that property, so the pattern you created is such an eigenvector.

Since this works with any single button press in the top row, it will also work with any combination of buttons on the top row. Similar patterns also work on a square board when working modulo 3, such as the Lights Out These patterns can of course be used to tile a rectangular board in the usual way.

In [GK2] it is proved using Fibbonaci polynomials that these are the only ones on rectangular grids. In that paper these patterns are called "Even Open Dominating Sets". The Orbix game type 1 also has eigenpatterns. There are 2 6 of them, generated by six of the patterns in which one hemisphere is lit.

Lets also consider button patterns for the standard Lights Out game where the affected lights are exactly those that you did NOT press. This means that:. The buttons along one diagonal form a quiet pattern in this non-reflexive game, so the solvability test shows that this game is not solvable when n is odd.

When n is even it turns out to be solvable, and from the 2 n quiet patterns we then find that there are 2 n such button patterns that in the normal square Lights-Out change only the lights that are not pressed. One general solution is shown below, and it is easily seen that it can be extended indefinitely to any even square by adding more L-shaped pieces.

The patterns that light up everything are generated by opposite pairs of buttons. Thus the patterns on the Orbix that change exactly the unpressed lights are the 2 6 patterns consisting of antipodal button pairs.

There are some Lights Out variants that as far as I know only exist in software form. Some of these offer some interesting insights. In Lights Out variants with more than two colours, repeated button presses usually cycle the affected lights through all possible states.

One unusual variant that does not is TileToggle which has 4 colours and two types of move. Move type 1 swaps white with red, and black with blue. Move type 2 swaps white with blue, and black with red. These two move types still commute, it does not matter which order they are done.

It turns out that this variant is equivalent to solving standard lights out twice. First treat white and black as if they were one colour, with red and blue being the second colour. You can then solve this like the standard 2-state game until all the lights are white or black. By doing a type 1 move combined with a type 2 move you swap the colours white and black. Using this combined move, you can then solve the black and white board.

This is very similar to how a standard 4-colour Lights Out game could be solved, by first using single moves to eliminate colours 1 and 3, and then using double moves to change any lights of colour 2 to colour 0. A common Lights Out type of game has moves where pressing a light on a rectangular grid changes all the lights in the same column and all the lights in the same row, including the light that was pushed.

The best known of these is Alien Tiles, which uses 4 colours and this move shape. Often the goal is to try to make all the lights the same colour, without actually prescribing which colour that should be.

Let s be the number of colours that the game has. If you choose four lights that lie in a rectangle shape i. This allows us to do a kind of light chasing on all but the first row and column. This solution does not always work for every pattern on every board. In particular, if w is the width of the board then in step d1 we need w-1 to be coprime to s, so that repeating that step will eventually make the first light of the row equal to the rest regardless of their initial state.

Step f also depends on this. Similarly h-1 where h is the height of the board must be coprime to s for step h and steps i2 and k to always work. If any of these steps do not work i. For the puzzle to be solvable in any colour, we need the last step to cycle through all possibilities. There is another way to look at this problem. If a board is always solvable then there is a unique way to change only a single light. Suppose you have this unique button pattern that changes only a single light.

You can permute the rows not containing that changing light and it still has the same effect, so these rows of the button pattern must be the same. Similarly the columns other than the one containing the changing light must all be the same. Let A denote the light itself. Let B denote all the other lights in the same row as A. Let C denote all the other lights in the same column as A. Let D denote all the remaining lights not in A, B, or C.

The argument above shows that in each of these four sets of lights, all the lights need to be pushed the same number of times. Let a, b, c, and d be those numbers. We can do this by combining the button pattern with the press-everything-pattern s-d times.

This simplified pattern no longer only changes a single light, but instead it just changes that light relative to the rest of the board. For this to be the kind of pattern we are looking for, the effects on D, C, and B must be the same modulo s. As mentioned before, this only works if h-1 and w-1 are coprime to s, so that they have inverses. The above analysis heavily relies on it being played on a rectangular board. It can also be played on other boards, for example Tacoyaki is played on the diagonals of a square.

It may be that the only way to fully solve it is to use the standard matrix techniques, i. Main Page. Merlin Number of positions, original Solution for original Merlin Number of positions, reissue Solution for reissued Merlin.

References The basic linear algebra The pseudo inverse Solving in the minimal number of moves. Some general definitions. The solvability test. Pressing only lit buttons. The Toggle Game. Results for the standard game on a torus. Results for the XL knight's game. Lights Out