# Zoki Mathematics – Workbook

## INTRODUCTION TO ZOKI DECK

Zoki deck is set of 34 cards. All sides have one of four indicia. First indicium is blank side, second is single blue stripe, third is pair of red stripes and the fourth is set of three black stripes. Total number of cards that can be made with 4 indicia is 70, but Zoki deck consists only with cards that have at least one pair of same indicia on opposite sides. That reduces number of used cards to 34.

Here is the complete set of Zoki deck:

Numbers written under cards represents total number of stripes on each card and can be called card value. Lowest number is zero (all 4 blank sides) and the highest value can be 12 (three black stripes on each side). Note that there is more than one Zoki card of some values.

We can use two ways for nomenclature Zoki cards:

• Values and Colors: All Zoki cards have its value and used colors on it. For example, Zoki 4 with all red colors on it should be nomenclature as 4R, Zoki 4 with one red and two blue signs should be nomenclature as 4BR, Zoki 4 with one black and one blue sign should be nomenclature as 4KB (K – blacK). Note that there are no Zokis of same values using same color combination!
• Single side values: We can also use single side values to nomenclature Zoki cards. Like in previous example, Zoki 4 with all red colors on it should be nomenclature as 2020 (two stripes – blank – two stripes – blank), Zoki 4 with one red and two blue stripes should be nomenclature as 2101 (two stripes – one stripe – blank – one stripe).

Why there are exactly 34 cards? What if 5th indicium is used?

Number of cards in deck is specified by simple mathematical formula: (N3+N):2 where N is number of indicia used in deck. So, in standard Zoki deck there are (43+4):2=34 cards. In expanded Zoki deck, there is 5th indicium, 4 green stripes. In that deck there are (53+5):2=65 cards.

## ZOKI 9 PUZZLE

Zoki 9 is an excellent solitaire game that allows you to play against the clock or gives you a chance to compete against others in order to complete the puzzle in the fastest time. There are no points to be calculated or totaled, and no scores that have to be kept. It is just mind against mind, memory against memory, card knowledge against card knowledge. This brainteaser is truly one of the greatest solitaire games in the world, because as opposed to most solitaire games, which will not allow you to finish a game every time, the completion rate for Zoki 9 is more than 99%. While some solitaire games’ probability of completion is even less than 50%, Zoki 9 has more combinations than any game in the world of its kind.

As we said, Zoki 9 puzzle is not 100% solvable even we are using only 27 of 34 Zoki cards. With all 34 Zokis possibility of solving puzzle is much less.

How do we choose Zoki cards for playing Zoki 9? We started with set of 12 Zokis that have 3 different signs (colors) on them:

 Statistic for these 12 Zokis Total of available dealings 220 Unsolvable dealings 0 Percentage of the solvable dealings 100% Probability having unsolvable 9 Zokis 0

Because there was low number of different dealings, adding Zokis that have same 3 colors, we now got a deck of 24 Zokis. We came to the possibility of not solving the puzzle. The statistic follows:

 Statistic for these 24 Zokis Total of available dealings 1.307.504 Unsolvable dealings 3.888 Percentage of the solvable dealings 99.703% Probability having unsolvable 9 Zokis 1:336

This percentage of solving the puzzle was extremely high and became completely acceptable. However, we needed three more Zokis so three players could play this game. Adding three more Zokis we got the following results:

 Statistic for these 27 Zokis Total of available dealings 4.686.825 Unsolvable dealings 39.357 Percentage of the solvable dealings 99.16% Probability having unsolvable 9 Zokis 1:119

Percentage we got was more then satisfactory, giving minimal almost theoretical chances, to have in dealing Zokis that couldn’t be solved, and it remained acceptable. Off course, the possibility of gaining 100% solvable 9 Zokis were explored. We started with the 24 deck of Zokis, where the solvability was 99.703%.

 After discarding the following 4 Zokis, each having equaled number of colors, we got the following results for the remaining 20 Zokis: Total of available dealings 167.960 Unsolvable dealings 40 Percentage of the solvable dealings 99.976% Probability having unsolvable 9 Zokis 1:4199

This result still hasn’t reached the 100%, and with calculated discard of any of two Zokis from the lower half, that have all 4 colors (for example 2000 and 3313) we gain 18 Zokis that are 100% solvable, which is enough for two players to play.

 After discarding the following 2 Zokis (as the above mentioned example), we have 18 Zokis that have 100% solvability: Total of available dealings 48.620 Unsolvable dealings 0 Percentage of the solvable dealings 100% Probability having unsolvable 9 Zokis 0

In the rules for the Zoki 9 game we put the deck consisting of 27 Zokis, as most optimal number of Zokis, and up to three players as well as for the difference in dealings (over 4.500.000 combinations) allowing minimal probability gaining unsolvable 9 Zokis with the odds 1:119. However, we have to emphasize that computer games Zoki 9 and Zoki 16 have 100% solvability.

How to quickly find out are 9 Zoki cards dealt solvable or not?

Easiest way is to look if there is one Zoki cards with more than one color on it and with no that color on other 8 cards. This is the most often reason for making Zoki 9 unsolvable. Because Zoki cards must have at least one pair of same signs on opposite sides, we can conclude that one of it sides must be in interior of 3×3 puzzle and there is no other same colors on other 8 cards to be paired with. Here are sample 9 Zoki cards that have no solves:

As we can see, only one Zoki cards have black stripes where at least one of them should be in interior of Zoki 9 puzzle and there are no more blacks on other cards to be paired with.

## ZOKI 16 PUZZLE

When playing Zoki 16, you will discover just how good your memory and patience really are. You will also see the importance of remembering and recognizing the Zokis and associating the stripes and numbers of each Zoki. Just like Zoki 9, Zoki 16 will give you hours of self-competition, sometimes frustrating, but nevertheless a mind challenging game full of entertainment. Zoki will sharpen the wits of adults, and challenge children’s minds as well.

Zoki 16 is one of the most solvable puzzles ever created. There are possible 2.203.961.430 different dealings and in only 2.204 dealings it was not possible to solve the 4×4 puzzle, which is only 0.0001%!! Or easy said the probability to deal 16 Zokis that are not solvable is exactly 1:999.983 and that means that it is quite impossible. Possibility having a “Flash Royal” in poker from the first hand dealt is 1:649.740, meaning that it is easier to gain a “Flash Royal” then 16 Zokis that can not solve the puzzle! This can be compared with Yachts dices: it is 4 times easier to get 7 dices with 7 same numbers (symbols) in one throw then getting 16 unsolvable Zokis! However, we have to mention that discarding Zoki card 0000 (also called “white snow”) and Zoki card 3333 (“black widow”), the number of absolving combinations goes even lower, almost 1:10.000.000!!

## DOMINATOR GAME

If you enjoy playing Dominos, then you are going to love playing Dominator. Dominator gives you all the fun of Dominos, but adds new dimensions to a very old game. Just as you do with the bones, you still connect the sides together and try to score points. Dominator challenges you to improve your play by shrinking the playing area.

How to choose the best card to make a move?

As described in game rules, main goal is to make table sum be divisible by 5. We will describe ways to recognize good Zokis in hand against “bad” Zokis when connecting only one side to table Zoki cards. While knowing that table sum would be often divisible by 5, easiest way for finding Zokis that would remain same sum or enlarge it for 5 points is to find Zoki card where the sum difference of 3 sides and fourth side is 5 and we are connecting that fourth side to table. For example, if Zoki card 2313 is connecting with its’ red side (2) and other three sides remain unconnected (3+1+3=7) calculation is simple:

NewTableSum = OldTableSum – 2 + 7 = OldTableSum + 5

That means that we subtract one red pair on table by connecting with red pair on card played and adding 7 stripes more which increases table sum by 5.

If you are connecting Zoki card with one of its’ same opposite sides, table sum will be increased by sum of other two sides on Zoki card. For example, if we are connecting Zoki 3121 on one of its’ blue sides (1) table sum will be increased by sum of other two sides (3+2=5).

For exercise:
Find one of these Zoki cards that cannot increase table sum by 5 in any way by connecting one of its sides.

Normally, table sum is not always been divisible by 5. So we should not only memorize Zoki cards that are “good” for increasing table sum by 5 but we have to learn main concept of finding Zoki whose sum difference between 3 sides and the fourth which we connect to table will make table sum divisible by 5.

For exercise:
Here is sample of playing Dominator. Table sum is currently 27. Find one of 8 offered Zokis that can make table sum divisible by 5.

When connecting two sides at same time mathematics is slightly different. New table sum will be increased (or decreased) by difference of two unconnected sides with two connected sides. For example, if we connect Zoki card 3121 on its red and blue side (2 and 1) we have remained unconnected sides 3 and 1. New table sum will be then:

NewTableSum = OldTableSum + (3 +1) – (2 + 1) = OldTableSum + 1

Tip: when we connect Zoki card that is two-symmetrical (e.g. Zoki card 2121, 3333, 1010) with two of its sides table sum remain unchanged!

For exercise:
Here is same sample of playing Dominator as in previous exercise. Table sum is currently 27. Find 3 of 8 offered Zokis that can make table sum divisible by 5 by connecting two sides.

There is one special situation in Dominator game. We call it “hole”. There is possibility of creating hole which is rounded with 4 Zoki cards. In that situation all stripes in hole that are “free” (not connected) are NOT counted to table sum. This is good moment if we have no other moves to make table sum divisible, to make a move with some “bad” Zoki card to make a hole and get table points. Following diagram will explain how you can count new table sum after creating “hole”:

As we can see at picture left, table sum is 27. When we want to put card at place that creates “hole” first we subtract all new inner stripes (marked with red circle). We got 1+3+3+2=9 inner stripes, so 27-9=18. Stripes on new placed Zoki that will be connected or to be in hole will not count at all. We have to find a Zoki card that can connect on side to make a hole (red stripes in sample) and with adding “outer” stripe to get table sum divisible by 5 (marked with green circle in sample). We added in sample Zoki 2020, where one pair of 0 and 2 sides remains “inner” and one pair of 0 and 2 becomes “outer” stripes. Finally, 18+2+0=20 points and we receive those points.

For exercise:
Are there more Zokis that can be connected like sample above that are not already on table to make table sum divisible by 5. If there, how many can be placed?

## YUGO 9 GAME

YuGo is destined to become one of the most popular games in the world. Like most two player games, YuGo will bring out the competitiveness as well as the ability to develop strategy in players. This overwhelming strategically based game will give you hours upon hours of brain challenging entertainment. It is a game of experience in whole-brain thinking, speed combining and mind challenging.

• YuGo is a type of intellectual games where the outcome is determined largely by rational choices among possible courses of action, similar with chess or checkers.
• This game requires us to apply a logical rule to a spatial array of symbols to find a best solution for move.
• Aim is to play as much Zoki cards from hand as possible and making more points while trying to prevent opponent to make a move.
• Like chess or checkers, players can see all opponents’ figures that are cards in YuGo, so player can try to predict opponents’ moves: with what card (figure) and on which place.
• Specifically, unlike chess and checkers, Zoki cards can be rotated so that is other thing players have to think about. Not just what card opponent could move and where, than in which rotation, too.
• In YuGo game both players have equal rights. Both are dealing three times, both are playing first three times unlike chess or checkers where one player is in advantage with his first move.
• The YuGo Game requires us to think both logically (left brain) and holistically – in parallel – to pick the information out of the spatial array of symbols (right brain).

Let’s make some statistical analyses. At the first move, one of players can play one Zoki card in the middle of playing field. So, there are 5 possible first moves for first player. Second player can play about 10-15 different moves on his turn. That is about 50-75 possible moves in just first move! As we know, in chess there are 20 different moves that players can play first. If we shorten list of this moves to ~10 logical moves, there are about 100 variations of first move in chess. Main difference of YuGo and Chess is that in chess, players have first “contact” after 5-10 moves, but in YuGo, round is over after at most 4-5 moves! That is one of reasons why YuGo game can be compared with Chess.

## PLUS-MINUS GAME

Plus-Minus is fun and educational game for 2 to 6 players! Children can practice their basic math skills through this perfect Zoki game!

The aim is not to be player who reaches point limit. That player looses the game. Point limit is calculated by simple formula:

PointLimit = 11 x NumberOfPlayers

So, for 2 players point limit is 22, for 3 players point limit is 33, etc. In this game you should practice your knowledge acquired in Dominator game. As you can see in game rules, for every move, value of one side can be subtracted from main table score while other three sides are adding.

Are there “bad” and “good” Zoki cards in Plus-Minus?

Of course there are. And we will learn which Zoki cards are good and which are bad. As you know the worse Zoki cards are cards that increases table sum even we subtract one of their sides.

Most points table sum will be increased when subtracting value of one side of Zoki card 3333 (also known as “Black widow”). This card will increase table sum by 6 (3+3+3-3=6). Normally, if we do not subtract one side, table sum will be increased for all 12 points. So this is the worst Zoki card in Plus-Minus game.

There are 21 Zoki card that increases table sum even one of it’s’ side is subtracted. We have left 6 Zoki cards that leave table sum unchanged and 7 Zoki cards that decreases table sum. Which are better?

Of course, better cards are ones that leave score unchanged forcing opponents not to get rid of their bad Zoki cards. If we play card that decreases table sum, next player can play his bad Zoki card that increases table sum.

Can players draw and play all cards from the deck while table score doesn’t reach point limit?

No, only if players always subtract the highest side value. Of course, players will not subtract one side value if that card with its total sum doesn’t reach point limit.

There are 34 Zoki cards. There are total of 34 each indicia on all cards. Sum of all indicia is 34 x (3+2+1+0) = 204.

We will use theory that players will always subtract the highest card value.

Each indicium is present on 19 Zoki cards. So there are total 19 Zoki cards where can be subtracted black color. That is 19×3=57. We have left 15 Zoki cards. 10 of them have red stripes. That would be maximum subtraction 2×10=20 points. On 5 left Zokis there are 4 with blue stripe on it. That is maximum 4 points subtractions. So, at most 57+20+4=81 points can be subtracted. If we know that there are total 204 stripes, points that are added are 204-81-123 points. This theoretical table sum will be, after all adding and subtracting 123-81=42 points which are enough to finish round for three players (44 is limit for 4 players). But, as we already said, players will not always subtract points from sum, because there are no reasons for that and they will not play those unreasonable moves. It is same as question from chess: Can be black player be in mate after only two moves? Answer is: yes, but players will not move those unreasonable moves.

## ZOKI MATCH

Zoki Match is a very simple, fast paced, eye-brain coordination game that is to be played by 2 or more players using 1 deck of Zoki cards. Those that have the natural ability to recognize objects very quickly will do very well in Zoki Match. This is another Zoki creation that will be great for family entertainment. It will improve your child’s, as well as your own, recognition skills and build self confidence in them as they succeed in the game.

Are there differences when making quickly decisions what Zoki cards to monitor for Match?

Yes, of course. As we know, there are three types of Zoki Match: Flush Match set of 3 cards with same values, Straight Match, set of 3 cards with values in row and Suite Match, set of Zoki cards with same 3 sides on each card.

 Flush Match As we can se in Zoki deck, there are values that cannot make set of 3 same values: 0, 1, 2, 10, 11 and 12. Values 3 and 9 have exactly one possible match, and values from 4 to 8, inclusive, can have 4 different matches, each. So, Zoki cards with values 4 to 8 are best for searching Flush Match. Straight Match Looking again at Zoki deck we can see that all cards can be used in Straight Matches, but with different probability. Matches [0, 1, 2] and [10, 11, 12] have only 2 possible matches each, values [1, 2, 3] and [9, 10, 11] have total 6 possible matches each, values [2, 3, 4] and [8, 9, 10] have total 24 possible matches each, values [3, 4, 5] and [7, 8, 9] have total 48 possible matches each and values [4, 5, 6], [5, 6, 7] and [6, 7, 8] have total 64 possible matches each. So, when looking for Straight Match best is for searching values between 4 and 8, as for Flush Match. Suite Match Zoki Suite Match is hardest to find. And there are a bit differences between searching for Suite Match. We can classify 3 groups of Zoki cards: Zoki cards with all same values (there are 4 cards in deck), Zoki cards with two colors on them (there are 18 such cards) and Zoki cards wit three colors on them (there are 12 such cards and they’re also named “cripple Zokis”). When Zoki card with all colors on it is watched, there are at most 3 possible Zoki cards that make a set with that card. So it is hardest way to find suite match with such Zoki card. When we look at two-colored Zoki card, there are 6 more Zoki cards that can make a Suite Match with such Zoki card. When look at cripple Zoki card, there is the most chance to get a Suite Match because there is 8 more Zokis that can make a Suite Match with such Zoki card.