Counting is the process of determining the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.

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1. Most counting systems use a -1, 0 and +1 system which gives each card in the deck a value. Usually cards 2-6 are +1, cards 7, 8 and 9 are 0 (neutral) and face cards and aces are -1. As the cards are dealt, the player keeps track of the count using this system to get an overall picture for the remaining cards in the shoe.
2. Counting is the process of determining the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set.
3. Card Counting Systems Ace-Five. If you’re interested in learning how to card count, but aren’t sure you want to learn a complex system right away, the Ace-Five Count might be right for you. It’s also perfect for casual players that want to cut into the house edge and have the possibility of ending up slightly ahead of the casino, but aren’t worried about playing blackjack for a living or squeezing out as big an advantage.

Counting sometimes involves numbers other than one; for example, when counting money, counting out change, 'counting by twos' (2, 4, 6, 8, 10, 12, ...), or 'counting by fives' (5, 10, 15, 20, 25, ...).

Practice a few different ones to see which works best for you, then put your new skills to the test playing blackjack. Card counting is a system for tracking the probability of receiving cards.

There is archaeological evidence suggesting that humans have been counting for at least 50,000 years.^[1] Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is, accountancy). Notched bones were also found in the Border Caves in South Africa that may suggest that the concept of counting was known to humans as far back as 44,000 BCE.^[2] The development of counting led to the development of mathematical notation, numeral systems, and writing.

Forms of counting[edit]

Counting can occur in a variety of forms.

Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time.

Counting can also be in the form of tally marks, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting; normal counting is done in base 10. Computers use base 2 counting (0s and 1s), also known as Boolean algebra.

Counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. Finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10 (unless you start in with your toes). Older finger counting used the four fingers and the three bones in each finger (phalanges) to count to the number twelve.^[3] Other hand-gesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. By using finger binary (base 2 counting), it is possible to keep a finger count up to 1023 = 2¹⁰ − 1.

Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.

Inclusive counting[edit]

Inclusive counting is usually encountered when dealing with time in the Romance languages.^[4] In exclusive counting languages such as English, when counting '8' days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting 'inclusively,' the Sunday (the start day) will be day 1 and therefore the following Sunday will be the eighth day. For example, the French phrase for 'fortnight' is quinzaine (15 [days]), and similar words are present in Greek (δεκαπενθήμερο, dekapenthímero), Spanish (quincena) and Portuguese (quinzena). In contrast, the English word 'fortnight' itself derives from 'a fourteen-night', as the archaic 'sennight' does from 'a seven-night'; the English words are not examples of inclusive counting.

Names based on inclusive counting appear in other calendars as well: in the Roman calendar the nones (meaning 'nine') is 8 days before the ides; and in the Christian calendar Quinquagesima (meaning 50) is 49 days before Easter Sunday.

Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc., and going up seven notes is an octave.

Education and development[edit]

Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's very first step into mathematics, and constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback do not count,^[5][6] and their languages do not have number words.

Many children at just 2 years of age have some skill in reciting the count list (that is, saying 'one, two, three, ...'). They can also answer questions of ordinality for small numbers, for example, 'What comes after three?'. They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set.^[7] Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed.^[8][9] In the meantime, children learn how to name cardinalities that they can subitize.

Counting in mathematics[edit]

In mathematics, the essence of counting a set and finding a result n, is that it establishes a one-to-one correspondence (or bijection) of the set with the set of numbers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematical induction, is that no bijection can exist between {1, 2, ..., n} and {1, 2, ..., m} unless n = m; this fact (together with the fact that two bijections can be composed to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics—hence (finite) combinatorics is sometimes referred to as 'the mathematics of counting.'

Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for anynatural numbern; these are called infinite sets, while those sets for which such a bijection does exist (for some n) are called finite sets. Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets.

The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called 'countably infinite.' This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all integers (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of real numbers, that can be shown to be 'too large' to admit a bijection with the natural numbers, and these sets are called 'uncountable.' Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities).

Counting, mostly of finite sets, has various applications in mathematics. One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map f: X → Y is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example.^{[citation needed]}

The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of permutations of {1, 2, ..., n} for any natural number n.

See also[edit]

• Yan tan tethera (Counting sheep in Britain)

References[edit]

1. ^An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990) p.9
2. ^'Early Human Counting Tools'. Math Timeline. Retrieved 2018-04-26.
3. ^Macey, Samuel L. (1989). The Dynamics of Progress: Time, Method, and Measure. Atlanta, Georgia: University of Georgia Press. p. 92. ISBN978-0-8203-3796-8.
4. ^James Evans, The History and Practice of Ancient Astronomy. Oxford University Press, 1998. ISBN019987445X. Chapter 4, page 164.
5. ^Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences, 105(35), 13179–13184.
6. ^Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.
7. ^Fuson, K.C. (1988). Children's counting and concepts of number. New York: Springer–Verlag.
8. ^Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.
9. ^Le Corre, M., Van de Walle, G., Brannon, E. M., Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52(2), 130–169.

Romes

Thanks for the replies kewlj. I play mostly double deck, with $10 to 25 mins, spreading to $75- $150.
The wild swings possible in the game make me pretty nervous (hence the spreads erroring on the conservative side). I'm hoping one way to smooth out the fluctuations is by using a more accurate counting method.

Your counting method might net you some more 'change' per hour, but the variance you're going to inevitably see in the game doesn't care about your counting method. Standard Deviation = 1.1*AvgBet. Standard Deviation for any number of hands = OriginalSD*Sqrt(NumHands). Notice these have absolutely nothing to do with your counting system but more your spread (affecting your AvgBet), which inevitably needs to be a certain amount to be profitable, regardless of the system you use.
If you're thinking about switching to lessen the fluctuations, I wouldn't switch... Just my opinion.

If you're thinking about switching to lessen the fluctuations, I wouldn't switch... Just my opinion.

I completely concur.
I have been involved in more 'which count' debate/discussions than any other topic on blackjack sites that I am on. This is a hugely polarizing topic, with knowledgeable and successful players deeply dug in on both sides of the debate. I am a proponent of simplicity as I honestly believe Hi-lo or K-O or another similar level one count (not A-5 or speed count) is more than adequate for 99% of players, including players that play professionally, like myself (most professional players and teams that I know or know of use hi-lo).
But in reality, it is not so much that I am a proponent of hi-lo, which has served me well, it is more that I am an opponent of the false and misleading information out there, influencing players to switch counts in search of unrealistic expectations and gains in profit or reduction in variance and swings. Claims made by proponents of higher level counts are just not realistic and often they will go as far as to cherry pick data and use flawed simulations to exaggerate their points.
Just yesterday on another site, one of the proponent guys used a sim, to compare two different counts, or in this case it was actually a variation of a single count, Hi-lo vs hi-lo lite (hi-lo lite is fewer and rounded index plays). Honest simulations show the difference in these approaches to be negligible. A difference of pennies. But this guy's simulation showed a difference of 15%, which would be significant. He then revealed that he used different betting ramps, betting more money at the same advantage of the position that he wanted to show in a favorable lite than the other position. Give me a break! That's apples to oranges, in no way a fair comparison.
Another commonly used tactic by the proponents of higher level counts is refusing to acknowledge the existence of higher error rates with higher level counts, involving adding or subtracting 3 or 4 numbers as opposed to 2 numbers, or side counting additional numbers. Scientific evidence confirms that the simpler the task the lower the error rate. As a task grows more difficult, even slightly, the error rate increases.
These people not only refuse to accept this standard recognized fact, but counter with things like 'with enough practice, I can play just as efficiently' or 'more complex tasks keep their mind sharper'. Sounds great, but it's just not factually true. If you are not going to account for some sort of higher error rate, which will reduce or eliminate any gain, you are not being honest as to real life expectations.
So, I am really a proponent of having accurate information for players, especially newer players to look at and not some cherry-picked data or information that will wrongly influence them. At that point if they choose to play a higher level count, that's fine and dandy. Just have reasonable expectations. For most players doing so won't be beneficial and for almost all players that currently know and play another count like hi-lo, of K-O efficiently, it will not be beneficial, as they will realize little if any difference in real world play.

Thanks again for the replies Romes and kewlj. I want to poke at the question of standard deviations one more time. I'm not an expert at the math, but intuitively I would think a level-two count system would help smooth out bankroll fluctuations.
Reasoning: with a level-two count system, your knowledge of the remaining deck composition is more nuanced. Therefore, when the higher-count method tells you that you have an advantage, it's more likely to be accurate. So, *when you place your big bets*, you're more likely to be placing them at a point when you have an advantage. In other words, big-bet losses are less likely with a higher count system. Which translates to less extreme fluctuations in your bankroll.
Is this reasoning sound?

Your main issue is you think as the count gets higher you're 'more likely to win'... or I suppose 'less likely to lose.' Not only are some of the differences in betting correlation negligible, but again each system tells you when you have an advantage, and that doesn't mean you'll win more / lose less FREQUENTLY. The dealer hand is just as likely to get blackjack as you. Where you make your money in the game is you get paid extra for blackjacks, the dealer does not. You can split and double to poor dealer cards, and in higher counts they will bust slightly more often. This helps you get paid more, NOT win more frequently. This is how you beat the game of blackjack... with blackjacks, doubles, splits, and dealer busts... not with a 'high count winning more often.'

Different Card Counting Systems App

Different counts don't effect the standard deviation of blackjack (to a worthwhile amount, in my opinion), and your EV (as discussed before) is: EV = AvgBet*NumHands*AvgEdge. With your Standard Deviation being SD = OriginalSD * Sqrt(NumHands) = (1.1*AvgBet) * Sqrt(NumHands).

Best Card Counting System

I again hope you see in these equations where the only variable you COULD be manipulating with a different counting system is the AverageBet... which if you have a more accurate system perhaps it tells you to bet more, more often... which would actually RAISE your SD's and fluctuations!

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Your level 2 and level 3 count systems sound 10x more sophisticated. I've done multiple counts, and they're quite similar all around. Other than being more error prone with a higher count, they really only get you pennies on the dollar more. Remember, just because you've 'more accurately' deciphered you have an advantage doesn't mean you're more likely to win the hand.

Is Counting Cards Hard