Latex notes

Latex notes

• The write notation for dagger is \dagger and not \dag. Check for instance, $\dagger$ and $\dag$ (This error is intended for illustration).
• Curly bracket, if intended to be visible, should be denoted by \{ and \}. For example, $\} \{$.
• Three dots $\cdots$ \cdots 

Random Math notes

Random Math notes

• Trace of any operator $A$ is preserved under unitary change of basis $U$ because in the new basis, $A' = U^{\dagger}AU$ and $Tr\{AB\}=Tr\{BA\}$ implies $Tr\{A'\}=Tr\{A\}$.
• Note that $e^{A+B}=e^A . e^ B$ only if A and B commute. If they don't commute, the RHS will have terms of the type $C_{n,r} A^{n-r}B^r$ while the similar terms on LHS will be of the form $A^{n-r}B^r + A^{n-r-1}BAB^{r-1} + \cdots + B^{r}A^{n-r}$. If the terms don't commute, then these similar terms on LHS and RHS are not really equivalent. If they commute, then they are equal.

How to solve Pyramid solitaire (work in progress)

How to solve Pyramid solitaire (work in progress)

There are 52 cards in the pyramid solitaire, out of which 28 are in visible in the form of a pyramid, while the remaining 24 are stacked together in an unknown order. So, by looking at the visible cards, we can know what is there in the stacked ones.

The version of Pyramid Solitaire that I am interested in right now is where it is possible to go through the stacked cards three times.

Now, A can annihilate Q, 2 can annihilate J and so on. In good interest, one wants to annihilate the cards already in the pyramid with each other, so that the stacked cards options don't get exhausted.

Here is the instruction to solve or find whether the card stack is solvable or not-

1. Make a list of A, 2, 3,... , Q and note how many of these cards are there in the pyramid.
2. If there are x A's and y Q's in the pyramid, then there has to be at least $z=x+y-4$ links between A and Q within the pyramid, which means that at least z number of pairs of A and Q must annihilate themselves within the pyramid itself. This restriction for all the other pairs puts an upper limit to the score in the solitaire as well as reducing the possible solution space to be scanned.
3. Note that it is possible for a pyramid to compulsorily have no links within if sum of each of its pairs is 4 and there are also 4 kings in the pack, which will add up to 28. Also note that if there are less than 4 kings, then at least one compulsory interlink exists.