Basics of Kenken Puzzles

If you don't know the rules of KenKen then you can refer to www.kenken.com for an excellent source of daily puzzles which also explains the rules. I'll summarise it as best I can, it's a puzzle where you need to fill in each cell such that each row and cell contains a unique sequence of numbers respectively. If you are more familiar with Sudoku, it's similar in terms of the fact that each row/column can only contain one 1, one 2, one 3.. one 9. In KenKen's case, you are allowed to use numbers 1,2...,n. Where n is the size of the puzzle. However, unlike Sudoku you must ensure each 'cage' (bolded subareas on the puzzle) adds, multiplies, subtracts or divides to the target number listed on the a top-left corner of the cage. Subtraction and Divisions can be done in any order, so 2 / 4 and 4 / 2 both match 2/ (division). However, the constraints of unique numbers must be adhered to at all times. For more information, just refer to www.kenken.com.

Today we will be solving and walking through a simple 6x6 KenKen puzzle (from www.kenken.com). There are several important concepts which will make your puzzle solving much simpler. KenKen can seem daunting at first, but it's a solvable puzzle without guessing! There is always a way to deduce/use logic to fill in the cells without trial and error. I hope to highlight some of these through this given KenKen puzzle:

So, how do we get started? We don't exactly have any free single 1x1 cage to immediately write down. Usually if there is a simple 1x1 cage you should write down those numbers first because they can give hints to surrounding cages in similar rows/columns.

Now note that today we are solving a 6x6 puzzle, hence we are only allowed to fill in numbers 1 to 6. This is important to note because it limits the number of possibilities for each cage. One of the first things I tend to look for first - is a cage in which I immediately know what numbers are contained in those cages but not necessarily the right order. For a 6x6 puzzle, you should look primarily for 11+ and 5- cages. Why is this? It's because there is only one possibility for each of the cages, namely 5 and 6, and 1 and 6 respectively. (Question: If it was a 8x8 puzzle, what numbers should you be looking for?) Don't worry too much about the ordering at the moment - the correct ordering will come as we fill the puzzle. Noting these down we get:

We can also note several one possibility cages, 15x = 3 * 5. There is no other way of multiplying two numbers less than 6 to equal 15. There's also another subtle one, 36x. How many possibilities does this have? Well it seems like there may be a few, 6*6*1, 6*3*2, 3*3*4. However, in our puzzle 36x exists in a row which means there can be no repeated terms in the factoring! Hence we can rule out 6*6*1 and 3*3*4 being valid possibilities for the 36x cage. This just leaves a single possibility, 6*3*2! Anymore cages that only have one possibility? 20x also looks tempting - let's see, 5*4*1 and 5*2*2. But wait! Again, 20x is on a row and can't have any repeated elements so it just leaves one possibility: 5*4*1! There may be more but let's just put these on the puzzle first:

You may note that we actually haven't even filled a single entry yet! This is quite normal, you'll slowly realise that many possibilities will start collapsing when you fill a few in!

Now we will employ a more tricky technique which is immensely useful in solving KenKen puzzles. In Mathematics, this is called the Pigeonhole Principle. Essentially, the Pigeonhole Principle states that if you have n items and wish to place them into m boxes with n > m, then at least 1 box will contain more than 1 item. How on earth are we going to apply this to the puzzle? We need to look in a vertical manner - note how the 3 rightmost columns (4th to 6th columns) already contain 4 possibilities of having a 5 in them? Wait! 4 possibilities in 3 columns that violates the rules of the puzzle! Hence, one of these must not belong in the 3 rightmost columns. Now look at the bottom-most 11+, the 5 in this cage can exist on the 3rd column! Therefore, if this puzzle has a solution the 5 must belong in the 3rd column. So let's fill this in...

Now the bottom-most 11+ cage must have 6 in the unfilled cell. Now you notice that it also fills in the other 11+ cage in the 3rd row! Because the number 6 exists in the 4th column, it means the 3rd row 11+ cage must have number 6 in the 5th column. This then cascades to 36x! The number 6 for 36x (2*3*6) must go to the rightmost column otherwise it will violate the other 2 columns that we just filled in. For Sudoku solvers this should be in your familiar territory! That one observation allowed us to chain-complete a portion of the puzzle:

What else can we fill in? It's important to consider new point of views after you have filled in a set of entries because by filling in correct positions for cages - it reduces the possibilities for cages that are related to it (relation comes from the single 1 to 6 constraint on the rows/columns). If we observe the 3/ (division) cage on the rightmost column, previously it has the possibility of 3/1, 6/2. However now that we have filled in 6 on the rightmost column, the only possibility is 3/1. If you jot this down on the cage, you notice that we can now fill in 15x because the number 3 in 3*5=15 cannot belong to the rightmost column anymore so hence we have uniquely determined the correct position. Again, a lot of cascading solutions - by solving the 15x, we now know the position of the 5 in 20x and the position in 3x (validate it for yourself). If you fill in the details you'll get something like the following:

Now it looks much better! At this stage, we can fill in 24x because we know the product of its divisors cannot contain a 3 as it will violate the first row only having one 3 (it's taken by the 15x). 24x = 1*4*6. Note that it also can't be 2*2*6 because it exists in a single row and we can't have duplicate elements because it'll violate the rules. Note that the 4- cage on the leftmost column can be uniquely identified to be 6-2. How did we do this?

There are actually multiple ways to view this - the simplest is to note that the first row possibilities are completely filled except for one number. We have used 1,4,6 for 24x and 3,5 for 15x. That leaves only the possibility of 2 in the left and uppermost cell. Now we can't put negative numbers so the only possibility for the other cell in the cage is 6 - making 6-2=4.

The other way to view this will prove to be more useful when solving larger KenKen puzzles. Note that on both row 1 and row 2 we have used the number 5 already. 4- only has two possibilities for a 6x6 KenKen puzzle, 6-2 and 5-1. Hence if 5 cannot go in either of the cells in the cage then it cannot be 5-1, leaving 6-2. Let's fill these in:

So now we have an interesting pattern. the 12x cage is a L-shaped figure. This actually provides useful information that you may not first realise. By filling in the 6 on the 2nd row, we know for certainty that the remainder of the 2nd row is numbers 2,3. Hence we can automatically deduce the 3rd row/3rd column cell to be 2 as shown above. Again, there are multiple ways to view this. We can view 12x as a cage and note that we have 2,3 inside the cage already. This leaves us the unique number remaining to be 2 (12 / (2*3)).

For any L-shaped cage, if there are a repeated element (in this case the number 2), then they will always go on the ends of the L-shape. The reason for this is simple, if they were placed any other way they would conflict and violate the unique number rule. Filling in these numbers yields the following partially complete puzzle:

There is another tip that is useful. If you look at the 4th column in which we have filled in (1,5,3,6). We can actually uniquely and correctly identify the 4th number in the first row by noting that we reduced 24x to 1*4*6. But 2 of these numbers have already been used in the 4th column, hence the missing and correct number to fill in is '4'. This gives the partially solved puzzle:

Give it a try to solve the remainder of the puzzle! Happy Solving! =)

[Spoilers Ahead]
Note:
If you get stuck, I've included the solution below:

2011 Trading - Part 6: The Step

As Confucius once said, “A journey of a thousand miles begins with a single step”. The year of 2011 was the year I took that the single step – but ultimately with thousands more to go.

2012 proves to be an equally interesting and if not challenging year! Happy New Year and best wishes to all!

2011 Trading - Part 5: The Paradox

It was closing in to the end of the year and there was a saying that never felt truer: ‘The more you know, the less you know’. I’ve had a taste of both fundamental and technical analysis – verdict? Incompleteness. With Roger’s inspiration to critique P/E ratios, I started critiquing the very tools I’ve begun to use!

Initially, Technical Analysis seemed to be an invaluable tool (and I think it still is) but the flaws and deceptions are becoming increasingly obvious. Simple concepts like support and resistance I undoubtedly keep using but the more ‘advanced’ and specialized concepts including Elliott Wave Theory, complex custom indicators (that were probably backtested and optimized to death), Fibonacci Retracements, Financial Astrology (W.D. Gann and the like) etc. seemed a bit far-fetched from reality.

The problem lies with perception and in particular trader bias, once you understand how people perceive events you can see faults in others and yourself more easily. For example, let’s say for Fibonacci retracement, which is used to determine downturn levels in which a particular share price will hit before it resumes its original trend – if the price rebounds close to or in particular on the a Fibonacci retracement level, this becomes positive reinforcement. You actually feel good about the trade and hence have a stronger association with the event – your mind actually self-reinforces that this type of technique indeed does work. Alright, let’s say if the price doesn’t rebound on a level – what would the trader think? One possibility might be, oh it just might hit one of the other (there’s a few) levels. The more levels or ‘area’ a particular indicator may have the greater the chance it’ll rebound on or close to a level on pure chance! Another possibility it might fall through completely, so the tool failed – but psychologically the trader is more biased to the happy/positive reinforcement trade than associate the tool to the negative reinforcement when the tool failed! This is a clear psychological flaw, and it was something I would never have considered if I didn’t learn Fundamental Analysis (as odd as it might sound) as well as learn about psychology in decision-making and perception! It’s amazing how multiple fields suddenly link up to get the ‘ah-ha’ moment.

Self-denial is one enemy that I feel has an immense impact on trading performance. Admittedly, I was in self-denial about many tools I was using – claiming I did analysis when I clearly did not (journals don’t lie, it’s easy to trick yourself to a reason in entering a trade – it’s harder to fool yourself into an impulse trade if you write down a logical reason), getting fooled by randomness (also the title of an excellent and highly recommended read by Taleb) by conjuring facts from pure random and coincidental events.

Ultimately you feel you have acquired less by knowing more – hence a paradox. It does make sense though, put in other (more confusing) words: “You only ever know what you don’t know when you know more about what you don’t know. If you don’t know much, you don’t know what you don’t know hence you feel like you know more than you know!”

2011 Trading - Part 4: The Fundamentals

It’s now reaching about the end of the third quarter of 2011. I was lucky to have a friend that generously lent me a book about fundamental analysis. To this point in time, I was Team Technical Analysis – he was my opposition, Team Fundamental Analysis. There are many arguments against Technical Analysis, one is that no trader ever gets stinking rich (i.e. to the tens of billions like Buffet) the other is more academic, the efficient market hypothesis. This hypothesis states that with the flow of information freely distributed, the market accounts for every information and thus all the currently known information is reflected into the current price. However, most people will know this is just bull – how could people make money consistently then? I’ll elaborate on this in a later section.

Anyways, I haven’t had much time (nor motivation) to learn about fundamentals. The book I was given is entitled Value.Able written by Roger Montgomery whom was able to write a surprisingly accessible book (I highly recommend it - especially if you trade Australian Equities as the examples are for once related to Australia).  Prior to reading the book, I’ve gone through a few chapters of Benjamin Graham’s titular fundamental analysis book, ‘The Intelligent Investor’ which I’ve found somewhat of a dry read but informative nevertheless. Value.able illustrates an excellent analogy on comparing a business to a bank. It discusses the importance of a high Return on Equity (ROE), the importance of economic moat (which is already an essential concept of FA) as well as arguments against holding debt and some clever ways of evaluating management performance. With these concepts, you can ‘value’ a company to what it’s truly worth (in terms of today) – if the current securities price is significantly less than what you evaluate the company to be (using his valuation model) then it’s a good buy.

What’s more surprising is his adamant stance against using P/E ratios, which is sometimes used as a silver bullet for Fundamentalists. I found this rebellious nature somewhat inspirational because this goes against what fund managers widely use as a benchmark. Surely the most highly paid professionals in the industry can’t have it wrong?

Unfortunately, with fundamental analysis there lies a series of problems. There’s a lot of uncertainty in evaluating what might happen in the future – the valuation method only discounts things that happen now – there are no certainties to economic conditions and trends that might happen in 10 years time. This difficulty is deemed as one of the downfalls for fundamental analysis, it’s amazingly hard to understand all the relationships correctly and then predict where trends will go in 2, 5 and let alone 10 years time! The best you can do is minimize the probability you get it wrong by sticking in industries you are well versed in which is recommended by Fundamentalists.

2011 Trading - Part 3: The Verdict

After racking up a series of losses and barely having any winning trades, I momentarily stopped trading to recover. I partially reflected on my current trading done to date and it just simply seemed like I didn't get any timely feedback on my trading style. It’s reported the best way to gain mastery of an ability is to get constant feedback so you know what’s right and what’s wrong, consistent practice and surprisingly a lack of do-or-die pressure. What I initially wanted to do was to emulate a do-or-die situation (you know, dramatic movies in which the hero/heroine survives and becomes insanely better! A man can dream…) by putting a large portion of my portfolio into equities and learn on the spot through pain and hardship!

Actually, it turns out only mostly experts perform better under pressure! Beginners become overwhelmed and overloaded with external pressure that they can’t concentrate on improving themselves and becoming better. Experts use their time-tested experience and leverage it with stress to learn faster – they simply perform better under pressure. Unsurprisingly, this also holds true with professional athletes.

This was one clear and distinct advantage of paper and simulated trading – you can identify key weaknesses before you trade with real money. You eliminate emotional stress of the ups and downs during profit/loss situations, and actually focus on learning the task at hand: learning how to trade correctly. This is more important than outcome! This concept is similar to Zen archery borrowed from Eastern Philosophy. (Refer to the book written by Eugen Herrigel (Zen in the Art of Archery) - who learns how to ‘hit the target’ without hitting it).

Luckily on the second half of the year, I started trading via CFDs, which provided an immensely better platform to conduct shorter-term trades and to identify key problems with my trading. The list I came up with emulated the following key problems:

• -        Profits were too small relative to the losses. Profits were taken too soon; losses were allowed to roam free.
• -        Lack of risk/reward analysis, most of the time the stop losses were larger than the potential rewards.
• -        Stop losses were sometimes ignored.
• -        Impatience and chasing the market, not wanting to wait for the next retracement.
• -        Lack of a clear long-term trading edge
• -        Poor money management and lack of position sizing relative to capital available
• -        Traded very short-term (1 minute charts – sometimes the movements are simply just due to noise)
• -        Lack of analysis through different timeframes, this contributed to trading against the general trend which generates a negative edge most of the time
• -        Emotional trading (take larger bets when losses are incurred to make up for them, however the profits were taken relatively quickly so if a large bet goes south – it makes things exponentially worse!)

Overall, it was a real eye-opener! I didn’t even consciously realise I had those faults until I started journaling the size of my trades, timing of my trades and (lack of) reason behind those trades. Of course, in the pass/fail verdict self-assessment I got a blazing F.