# Candyland

Further out, the squares that are accessible with double cards have different probabilities because of their different frequencies in the deck. With care it is possible to construct a transition matrix for the game which enumerates the probabilities of moving from every game state to every other game state.

All that is needed now, is to create a column identity vector with 1. The row vector output will show the probability distribution for where the player token could be after one card draw.

Below are the results represented in graphical form, with shading denoting the probabilities. Darker red representing high probabilities and Lighter red representing low probabilities. After one draw, the darkest areas are, obviously, the ones that are reached by the cards that have the highest distribution in the deck the single colours. The spaces obtained by the double colours are lighter shaded some lighter than others because there are less of these cards in the deck.

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## 20th Annual Candyland Event

If we multiply the above output vector by the transition matrix again, the new output vector represents the superposition of probabilities of where the player token could be after the second drawn card starting from all the weighted starting positions of the first vector. The result is shown below. The pink squares are the same shade, and there is still no way to reach the finish line with just two card draws.

After the third card draw, the probability distribution looks like the picture below. The clouds are getting wider, and more of the board is shaded. After four draws, and interesting event occurs — it is mathematically possible to win the game!

The probability of landing of finishing the game in just four moves is approximately once in every 10, games. There are multiple ways to achieve this victory, but all require that the first card drawn be the pink special card that takes the player to square In our version of the game, this is the Ice Cream. By five draws, there are more ways to achieve the victory condition as you can see from the probability increase of the token being on the finish space.

Below are thumbnails of the board showing the percentage of finishing increasing with each drawn card. Here is a short video animation showing the probability distribution for the first 50 card draws. Below is a graph of the probability of a single person finishing the game by move- n. The Modal number of card draws to is More games will finish using this number of moves than any other number.

The Median number of card draws to is Half the number of games will be shorter than this, and half longer. The Arithmetic Mean average number of card draws to is I was curious to learn just how close the replace-and-reshuffle simplification to the model was to real results, so I decided run an objective comparison. I created a Monte-Carlo simulation of the game so that I could automate the game and execute it millions of times. This did not take long to code, as most of the work in creating the first simulation was the data-structures, and serialization of state and this could all be re-used.

Another benefit of writing the simulation is that it is possible to model a game with more than one player. There are two reasons why this is important:. When there are more players, more cards are drawn, so it is much more likely that the deck will need to be reshuffled sometimes many times.

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Reshuffling creates the situation that the special pink cards are encountered multiple times. In real-life, a game ends when the first person crosses the finish line. With multiple players, there are multiple chances for the game to end in a round. They look pretty close!

This is a good thing, and shows that we've probably not made any simple logic errors in the code — Getting the same results using two different mechansims of calculation always makes one feel warm inside. The fact that the curves are pretty close also confirms that our simplification approximation was a valid one to make. The 'true' curve, the Monte-Carlo one, peaks a little higher at the mode, which is to be expected. Well, to complete the game in a small number of moves requires the use of one of the pink cards and so these will be used up. Until the deck is reshuffled post 64 drawn cards , there is zero chance of encountering that pink card again.

However, in the Markov Chain model , it is possible to encounter the same pink card again, and encountering the same card twice will send you backwards. This does not happen often, but it does enough to slightly lower the perentage chance at the modal peak. Another way to look at the data is to view the Cumulative Probability of a game finishing by at least n -moves. As you can see, the simplified Markov model seems to slightly over-estimate the probabilities once about half the deck has been drawn.

### But What Can We Do?

The Mode and Median of the two methods are the same at 22 and 25 respectively, but the Mean decreases from When there is more than one player in a game, the statistics change quite considerably because of the two reasons mentioned earlier. Below is the graph of the percentage chance of completing a game in n -rounds.

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Note — Here 'round' refers to each person taking a card, and this is subtly different from drawing cards. They always ask on the weekends and days off when they will get to come back to Candy land. The teachers are friendly and helpful. Overall my family and i Have had a great experience at Candy land. I'm not sure what the comment below is referring to but I personally know of two families which go there that I babysit for! The kids and parents both talk about how much fun this school is! There is two huge play yards that really allows children to be kids.

The school is amazing! Definitely go check it out. Also cleanliness. Go into any school around the area and you tell me if you find anything as clean! I pick these kids up all the time and I'm so impressed how clean it is! Candyland Academy in Edison, New Jersey , is an innovative children's learning center that provides for children ages 15 months through 13 years, and encourages a happy and supportive learning environment.

At our center, children will be encouraged to learn about friendship, sharing, following directions, and using their imagination.

## Candyland | North Campus Research Complex

The education, enrichment, and well-being of your child is our greatest priority. Candyland Academy in Edison, NJ provides a full schedule of activities that stimulate the mind and body, so your child will never have a dull moment. Our curriculum is always age-appropriate and ensures that your child receives both learning and fun with us. Offering plenty of play time, we have things to do for children of all ages.

We can even arrange full- or part-time care. Quality Toddler Care Early Education.

Children are introduced to a well-rounded curriculum that covers a variety of stimulating subjects such as art, music, reading, science, math, and dance. Flexible Schedules. The flexible schedules at Candyland Academy are offered in full- and half-day sessions.