Cryptarithms

                              

Cryptarithms or crypt-arithmetic problems are encrypted math problems, where numbers in a given mathematical expression are represented by letters or other symbols. 

There are two main types of cryptarythms: 

1. Hindu problems.
2. Alphametics. 

In Hindu problems, every digit in a mathematical expression is concealed with the same symbol, usually an asterisk. Surprisingly, despite the fact that once the original digits have been replaced by asterisks and they are indistinguishable, these problems can be solved.

Alphametics are puzzles where each digit is replaced by a different symbol, usually a letter. Many alphametics “spell out” words, making them more attractive and entertaining. One of the most famous was invented by Henry Ernest Dudeney, a British puzzlist, in 1924: SEND + MORE = MONEY.

By substituting S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2 the cryptarithm translates into: 9567+1085=10652. In the second half of the week, we’ll learn how this and other cryptarithms are solved.

A bit of history

Cryptarithms first appeared in the United States in 1864, but it is believed that they were invented much earlier, in Ancient China. These original cryptarithms were mainly of the “Hindu” type. In the early twentieth century, Simon Vatriquant, a Belgian mathematician pseudo-named ‘minos’, and mistakingly called Maurice Vatriquant, took them much more seriously. He published many cryptarithms in a mathematical journal called “Sphinx”, published during the first half of the twentieth century. Maurice Kraitchick, another well-known mathematician, was the editor of this journal. The journal was dedicated to math puzzles, a branch of mathematics that is known today as recreational mathematics. Jorge Soares, among others, has a great cryptarithm site dedicated to the Sphinx. If you are already well-versed in cryptarithms, you might want to challenge yourselves with some of the cryptarithms in his site.

Hindu problems originated in India in the middle ages and are sometimes referred to as arithmetical restorations or skeleton problems. In these problems, digits are replaced by asterisks, for example: * + * = 2.

Each asterisk represented one missing digit (from 0 and 9). Two or more asterisks in the same problem may represent different digits.

For example, for the problem * + * = 2 there are two possible solutions:

• 1 + 1 = 2 – in this solution each asterisk represents the digit 1
• 2 + 0 = 2 – in this solution one asterisk represents the digit 0 and the other asterisk represents the digit 2

Note that we are counting 2 + 0 and 0 + 2 as the same solution.

Remember the following points:

• Some puzzles might have several solutions and some might have no solution
• Two adjacent asterisks represent a 2-digit number, like 74. Three adjacent asterisks represent a 3-digit number, and so on. In these cases the leftmost asterisk cannot represent the number 0, i.e. it is illegal to write to represent the number 8 as 08.

Solve the following cryptarithm. There is only one solution!

    * * * × * – * * = 1

The solution is: 100 × 1 – 99 = 1 A 3-digit number multiplied by a 1-digit number will result in either a 3 or 4-digit answer, unless the 1-digit number is 0, in which case the multiplication result is 0. This will lead to an impossible problem: 0-**=1 since negative numbers are not allowed. So, we are dealing with a multiplication that will give either a 4 or 3-digit result. It cannot be a 4-digit result, because a 4-digit number minus a 2-digit number will result in a 3 or 4-digit number and not the number 1 as required. So, we now know that the result of the multiplication is a 3-digit number which, when we subtract from it a 2-digit number, we get ‘1’. The only possible solution is: 100-99=1 so the multiplication must be 100x1 and the whole solution: 100x1-99=1.

Solve the following cryptarithm. There is only one solution!

 * * – * = 99

The solution is: 99 - 0 = 99 Look at the question backwards: 99 + * = **. The only number you can add to 99 that will still give a 2-digit answer is 0. Hence, the only possible solution is 99 - 0 = 99

Solve the following cryptarithm. There is only one solution!

* * × * * = 169

The solution is:1 3 × 1 3 = 169. Here we have to use the fact that there is only one solution. Hence, both the multiplicands have to be prime numbers. Since 11 × 11 already gives a close answer (121 is close to 169), we can try the next Prime up - 13. Divide 169 by 13 to get 13 - and the problem is solved. Of course, if you happen to know that 13 squared is equal to 169 that does make things easier :-)

How many solutions are there to the following cryptarithm?
* * * × * – * * = 4

The answers is: 4 and solutions are,
100 × 1 – 96 = 4
101 × 1 – 97 = 4
102 × 1 – 98 = 4
103 × 1 – 99 = 4

Solve the following cryptarithm:
* * × * * – * = 120

The solutions are:

11 × 11 – 1 = 120

10 × 12 – 0 = 120

The way to work this one out is to ‘add’ the single asterix to the answer to get: * * × * * = 120 + * . Now check which numbers between 120 and 129 have two 2-digit multiplicands.

Let us move on to 

What are alphametics?

Alphametics are cryptarithms that spell out words. Given a mathematical expression, every digit in the expression is replaced by a letter. One of the most famous alphametics, spelling out ‘SEND MORE MONEY’ appears above. This alphametic was first published by Henry Dudeney, a British puzzlist, in 1924.

Five rules govern alphametics:

1. Identical digits are replaced by the same letter. Different digits are replaced by different letters.
2. After replacing all the letters with digits, the resulting arithmetic expression must be mathematically correct.
3. Numbers cannot start with 0. For example, the number 0900 is illegal.
4. Each problem must have exactly one solution, unless stated otherwise (unlike the “Hindu” problems where there might be no solution or multiple solutions).
5. The problems will be in base 10 unless otherwise specified. This means that the letters replace some or all of the 10 digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

I recommend reading this before watching the video… There are key elements to solving most alphametics.

• In many cases the result of an addition problem is one digit longer (in digit-length) than the addends - the numbers added. If there are only two addends, this implies that the extra digit is the number 1.

Let’s look at a very simple alphametic: ME+ME=BEE

The letter B must represent the digit 1, since when you add two 2-digit numbers you cannot possibly get a number larger than 198. That happens when both addends are 99. Since M and E are two different numbers, they will certainly be even smaller than 99! In any case, the hundreds digit in the sum, represented by B in our example, must be 1.

• In two addend alphametics, there may be columns that have the same letter in both the addends and the result. If such a column is the units column, that letter must be 0. Otherwise, it can either be 0 or 9 (and then there is a carry).

In the alphametic: ME+ME=BEE the column of the unit’s digits is: E+E=E There is only one digit, which has the property that when you add it to itself you get the same digit as the result – zero! Only the sum of two zeros is zero, so E must be equal to 0.

The solution to this alphametic is therefore: B=1, E=0, M=5: 50+50=100.

Here are some tips for solving more complicated alphametics.

• If there are more than 2 addends, the same rules apply but need to be adjusted to accommodate other possibilities. If there are four identical letters in the units column (one of them the sum), this letter can now be: 0 or 5 (because 5+5+5=15). If there are four identical letters in a different column (one of them the sum), this letter can now be: 0 or 5 (no carry), 4 or 9 (carry 2). Four identical letters in a column other than the units column means a 1 could not have been carried over (why not?). This rule can be worked out for more than 3 addends as well…

• It is wise to turn subtraction problems into addition problems by adding the result to the smaller addend to get the larger one.
• When faced with a few options for a letter, try one out until you either get the correct answer, or find a contradiction.

Now let’s look at a slightly more advanced cryptarithm. Above video shows how to solve the alphametic: NO + GUN + NO = HUNT. Note the ‘neat’ sentence: “No gun, no hunt!”

Solve the following alphametic. It might be helpful to rewrite the alphametic as a column addition problem.

MA + PA = MOO

The solution is: 15 + 85 = 100
M=1 (hundred’s digit)
P+1 or P+2 (if there is a carry from the first column) =10+O
P can either be 9, in which case O=0, or 8, in which case O=0. In any case, O=0, hence, A must be 5 and P=8.

Solve the following alphametic. It might be helpful to rewrite the alphametic as a column addition problem.

ON + TO = OFF

The solution is: 19 + 81 = 100
Explanation: O has to be 1 because the result is 3 digits long. The first digit of the result has to be 1 because the sum of two 2-digit numbers cannot be larger than 198 at the most.
Replacing all the O’s with 1 we now have: 1N+T1=1FF
If we write the problem in columns, both the first column and the second column carry over a 1 to the next column. For the second to the first this is obvious. From the first to the second this is because the result in both columns is the same: F, yet the addends are different - N and T. This means that both N+1 and T+1+1 are larger than 9.

Notice that:
N+1 = 10+F and T+1+1 = 10+F
So N = T + 1
The only possibilities left for N and T are: 5,6; 6,7; 7,8 or 8,9 which is in fact, the correct answer. N=9 and T=8. Substituting these back into the problem yields the value of F - 0.

Solve the following alphametic. It might be helpful to rewrite the alphametic as a column addition problem.

BAD + DAD = DORA

The solution is: 921 + 121 = 1042
The solution is: 921 + 121 = 1042
Explanation: D must be 1 since it is the first digit in the 4-digit result of a two 3-digit addend sum. Replacing all D’s with 1, we get in the first column 1+1=A. So, A must be 2. Replacing the A’s with 2 - we get in the second column 2+2=R, so R is 4. In the third column, B+1 = 10, so B=9.

Solve the following alphametic. It might be helpful to rewrite the alphametic as a column addition problem.

WOW + WOW + WOW + WOW + WOW = SNOW

The solution is: 575 + 575 + 575 + 575 + 575 = 2875
Explanation: Rewrite the problem as WOWx5=SNOW
It is obvious from this that W=5 since the last digit in the result of multiplication of any number by 5 is always 5. Replacing all W’s with 5, we get: 5O5x5=SNO5.
Since 5x5=25, Ox5+2 must equal O. The only options for O are: 2 or 7. Trial and error give us the solution.

It is more difficult to find multiplication cryptarithms that also spell out words. Still, alphametic multiplication (and division) problems are still fun! Try this one. There is no need to write it down as column multiplication: AB x AB = ABB

he solution is: 10 x 10 = 100
For BxB to equal B, B can only be 0,1,5 or 6.
If we look at the problem as AB squared = ABB, we know that ABB must be a 3-digit square number. That rules out B=1, B=5 and B=6, so B=0.
To give a 3 digit result, A can be either 1,2 or 3.
However for A0xA0=A00, the only possible answer is A=1.

Some multiplication rules can really help us when we get stuck on a cryptarithm. Here are some rules you should know before taking the ‘plunge’ into the more difficult problems…

• A number is divisible by 2 if it’s unit’s digit is an even number.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if the number ends with a 0 or a 5.
• A number is divisible by 6 if it’s an even number and the sum of its digits is divisible by 3.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 11 when the difference between the sum of the digits in the even positions (looking at the number from left to right) and the sum of the digits in the odd positions is divisible by 11.

Examples

The number 43242543 is divisible by 9 and by 3 because the digit-sum of the number: 4+3+2+4+2+5+4+3=27 is divisible both by 3 and by 9.

Is the number 1352467928 divisible by 11?

The sum of the digits in the even spots: 3+2+6+9+8=28

The sum of the digits in the odd spots: 1+5+4+7+2=19

Since 28-19=9 and 9 is not divisible by 11 the whole number is not divisible by 11.

We can ‘fix’ the number to be divisible by 11 if we can ‘fix’ the difference to be, i.e. 11. This can be done by adding 2 to one of the digits in the even spots. Let’s change the number 3 to 5. We get: 1552467928. This number is divisible by 11:

The sum of the digits in the even spots: 5+2+6+9+8=30

The sum of the digits in the odd spots: 1+5+4+7+2=19

Since 30-19=11 and 11 is divisible by 11 the whole number, 1552467928, is divisible by 11.

Another fun genre is when the cryptarithms spell special words. Here is an example that Truman Collins has on his website:http://www.tkcs-collins.com/truman/alphamet/alphamet.shtml

SATURN + URANUS = PLANETS.

There are three possible solutions for this cryptarithm.

Truman has a lot of similar examples, and you can even generate your own ‘related words’ cryptarithms with his online cryptarithm generator. http://www.cadaeic.net/alphas.htmOther literary cryptarithms can be found on Mike Keith’s excellent website, http://www.tkcs-collins.com/truman/alphamet/alpha_gen.shtml 

Try and solve the following cryptarithm: I x KINGJO = JOKING The hint is: I=4

Don’t Sell Yourself Cheap!!

The father said to his son: “You have graduated with distinction, this is a car I got many years ago. It great but old. 

Before I give it to you, I would like you to take it to the used-car shop and tell them you want to sell it.

His son went to the car store. Later he came back to his father and said: “They offered me $ 1,000 because it seemed too worn out.” 

The father said, “Fine take now to the auction store.” 

The son went to the auction store and came back to his father and said, “They offered me $ 100 because it's a very old car.” 

The father smiled and asked his son to take the car to Collective Car Club and asked him to show it to them.

The son took the car to the club, and came back and told his father, "Some people in the club offered me $ 100,000 since it is a rare Nissan Skyline R34, which is a very popular car and many have sought after." 

The father smiled again and said to his son, "The right place will always value you in the right way." If you are not of value, do not be angry...  this only means that you are in the wrong place. 

Those who know your worth are those who will value you correctly. 

Never stay in a place where someone does not see your value.

7 tips to keep you safe online

With people spending hours on the web and turning to online services for practically everything under the sun, it is an opportune time for hackers and data-stealing malware to run amuck. Consequently, many have been worried about keeping their private data safe and secure and away from those with malicious intentions. However, experts say there are simple steps you can take to minimize the risk and protect your data.

Back up your data, externally

Though often ignored or postponed, keeping copies of your digital data is one of the simplest and most sensible ways to protect it. Backing up data means that your information and documents will be safe if you were to lose any of your devices, or if they become damaged or stolen. Using an external hard drive, and in particular, one that supports hard drive encryption is a good place to start. It's also a good idea to scan important documents and save them on your computer and hard disk drive (HDD).

Change your passwords, regularly

Your existing passwords must be at least 15 characters long and changed frequently. More importantly, do not keep a note of your password on or near the device you are using.

Keep your birthday and address off the internet

If you have information such as your address or birthday on your social media, know that these can easily be used for malicious purposes. To avoid this risk, try not to declare your birthday or place of residence on online platforms.

Check your camera and Bluetooth, stat

Hackers can actually access your mobile and laptop cameras and record you, even if they are turned off. When your device is not in use, instead of leaving it on sleep mode, turn it off completely, and if looking for a low-tech solution cover it with some black tape. When it comes to Bluetooth, you must exercise more caution as it is a literal open door for pirates, so make sure it is turned off.

Don't use public WiFi

Your Wifi range often extends the boundaries of your house and signals can be picked up from near its perimeters, making them a target for hackers. The best precaution you can take will be to use a password consisting of a mix of letters, numbers and special characters. In addition, always try to avoid using public wireless networks and, if you have to make sure the site addresses start with HTTPS (with an SSL certificate) instead of HTTP.

Use trustworthy antivirus software

Malware, spyware and viruses spell big trouble for many users and devices. The best way to avoid them is to use good antivirus software that will provide reliable anti-malware protection.

Turn on your notifications and alerts

Despite all the precautions you take, fraudsters still may find a way to bypass the measures you take. In such cases, the sooner you discover the threat the better it is for recovery and minimizing further damage. Nowadays, many banks send free notifications and warnings when they detect unusual activities. So call your bank or log in to its app to see if you have your notifications turned on.

Lesson of time

“When a bird is alive, it eats ants.  When the bird is dead, ants eat the bird.  Time and circumstances can change at any time.  Don’t devalue or hurt anyone in life.

  You may be powerful today, but remember, time is more powerful than you.  One tree makes a million match sticks but only one match is needed to burn a million trees.  So be good and do good.” - Karmic saying

Time is more powerful than You

Moral: Time is powerful. You can be king one day and penny less other day. You should be polite.

COVID Risk

Source :Texas Medical Association

KFC Replica

Thoughts

1. Passengers on the bus

Visualize yourself driving a big red bus. There are passengers on the bus, and as you drive around, some get on and some get off.The passengers represent your thoughts. Now imagine yourself talking to them. This is a great way to become more mindful of your thinking, while at the same time, distancing yourself from your thoughts.

What you need to remember is that you are the driver of this bus, the one who calls the shots. The passengers are only temporary. They will come and go.By doing so, you can take control of the bus — your mind-bus — by saying things such as,

“Thank you for your feedback, but this is my bus,” or “Hey, this is your stop, time to get off.”

You can use this technique for any type of negative thinking, but research shows it is particularly effective for improving self-control.

2. Clouds in the sky

Imagine your thoughts as clouds floating through the sky. Sometimes they’re dark and angry, sometimes they’re light and calm. But you are not the clouds.

You are the blue sky who notices the clouds, without engaging. You simply observe them until they pass. This is the practice of self-observation, which means mindfully observing how you think.

Consider this example. If I asked you what you were thinking, you might notice that you’re kicking yourself over a missed opportunity, worrying about money, or calling yourself stupid. The idea is to take a step back and observe these thoughts until they will pass. The good news is — they will pass. Everything passes, good and bad.

When you practice this regularly, you will create a sense of detachment when challenging thoughts arise. More and more, you’ll realise you are not your thoughts, and instead of feeling overwhelmed, there will be a space, and you will be able to respond in a rational manner.

3. First and second darts

First darts are inescapable pains that life throws at us.It might be a tough breakup, a lost opportunity, or the death of a loved one. These unavoidable pains are the essence of human existence, and if you live and love, some of these will fall on your doorstep.In reality, however, most of our problems are not caused by first darts. They are caused by how we respond to them. 

Second darts are the darts we throw at ourselves.These are our reactions to first darts, and this is the source of much of our suffering.These second dart reactions are more common than you think.

How often have you argued with your boss, before you’ve even gotten out of the shower?

How many times have you brought the morning traffic into work?

How often have you brought work frustrations home for dinner?

This is the essence of suffering, secondary reactions to painful events, which are often more destructive than the original experience.

Next time you recognise first darts, instead of resisting them, you should accept them completely. If you do get stuck in traffic, or frustrated in work, accept it and move on because it’s our resistance to pain that causes our suffering.

Take away message

By holding the most truth in the least amount of space, metaphors can help you to cope with abstract psychological concepts such as overthinking.

Next time your mind is busy, you don’t have to feel overwhelmed. You could kick those troublesome passengers off the bus. You could observe those dark angry clouds as they float by — without engaging. Or you could accept those first darts before they turn into suffering.

You can’t stop thinking, no matter how hard you try, but you can distance yourself from problematic thoughts — then they won’t feel so loud.

Brassica oleracea aka Broccoli 🥦

It's no coincidence that more than 300 research studies on broccoli have converged in one unique area of health science—the development of cancer—and its relationship to three metabolic problems in the body. Those three problems are (1) chronic inflammation (2) oxidative stress, and (3) inadequate detoxification. While these types of problems have yet to become part of the public health spotlight, they are essential to understanding broccoli's unique health benefits. Over the past 10 years, research has made it clear that our risk of cancer in several different organ systems is related to the combination of these three problems.

The Cancer/Inflammation/Oxidative Stress/Detox Connection

In health science research, there is a growing body of evidence relating cancer risk to a series of environmental, dietary, and body system factors. Understanding this set of factors can be very helpful in making sense of broccoli and its health benefits.

Anti-Inflammatory Benefits of Broccoli

When threatened with dangerous levels of potential toxins, or dangerous numbers of overly-reactive, oxygen-containing molecules, signals are sent within our body to our inflammatory system, directing it to "kick in" and help protect our body from potential damage. One key signaling device is a molecule called Nf-kappaB. When faced with the type of dangers described above, the NF-kappaB signaling system is used to "rev up" our inflammatory response and increase production of inflammatory components (for example, IL-6, IL-1beta, TNF-alpha, iNOS and COX-2). This process works beautifully in temporary, short-term circumstances when healing from injury is required. When it continues indefinitely at a constant pace, however, it can put us at risk for serious health problems, including the development of cancer.

Isothiocyanates (ITCs) in Broccoli

Research studies have made it clear that the NF-kappaB signaling system that is used to "rev up" our inflammatory response can be significantly suppressed by isothiocyanates (ITCs). ITCs—the compounds made from glucosinolates found in broccoli and other cruciferous vegetables—actually help to shut down the genetic machinery used to produce NF-kappaB and other components of the inflammatory system. These anti-inflammatory benefits of ITCs have been clearly demonstrated in lab and animal studies. However, it can sometimes be tricky to translate the results of these lab and animal studies in practical take-away recommendations for everyday eating.

The primary anti-inflammatory ITC provided by broccoli is sulforaphane. This ITC can be directly produced from broccoli's glucoraphanin content. Numerous anti-inflammatory mechanisms for sulforaphane are well known, including inactivation of the NF-kappa B pathway. In this context, it is interesting to note that the predominance of sulforaphane in broccoli is limited to the heading version of this vegetable. Also widely enjoyed worldwide is "non-heading" broccoli, often called sprouting broccoli, broccoli raab, broccoli rabe, or rapini. In these non-heading varieties of broccoli, iberin is the most common ITC, and it is derived from glucoiberin, which is one of the more common glucosinolates in non-heading broccoli). Yet another anti-inflammatory compound present in both heading and non-heading varieties of broccoli is glucobrassicin. (And in this case the corresponding ITC derived from glucobrassicin is indole-3-carbinol.)

Omega-3s in Broccoli

Lack of omega-3 fat is dietary problem that can cause over-activation of the inflammatory system. The reason is simple: many key anti-inflammatory messaging molecules (like PGH3, TXA3, PGI3, and LTE5) are made from omega-3 fats. While we are not accustomed to thinking about non-fatty vegetables as sources of omega-3 fats, it would probably be a good idea for us to change our thinking in this area. While there are limited amounts of omega-3s in low-fat vegetables like broccoli, it is equally true that their levels of omega-3s can still play an important role in balancing our inflammatory system activity. In 100 calories' worth of broccoli (about 2 cups) there are approximately 400 milligrams of omega-3s (in the form of alpha-linolenic acid, or ALA). That amount of ALA falls into the same general ballpark as the amount provided by one soft gel capsule of flax oil. While we would not want to depend on broccoli as our sole source of dietary omega-3s, we still get important anti-inflammatory benefits from the omega-3s it provides.

Other Anti-Inflammatory Benefits of Broccoli

Broccoli is a rich source of one particular phytonutrient (a flavonol) called kaempferol. Especially inside of our digestive tract, kaempferol has the ability to lessen the impact of allergy-related substances (by lowering the immune system's production of IgE-antibodies). By lessening the impact of allergy-related substances, the kaempferol in broccoli can help lower our risk of chronic inflammation.

Broccoli's Antioxidant Benefits

Vitamins, minerals, and phytonutrients all contribute to the antioxidant benefits provided by our food. Broccoli is a premiere example of a vegetable providing all three types of antioxidants. In the vitamin category, among all 100 of our WHFoods, broccoli represents our 3rd best source of vitamin C,10th best source of vitamin E, and 16th best source of vitamin A (in the form of carotenoids). It also serves as our top source of chromium, a very good source of manganese, and a good source of selenium and zinc. But it is the phytonutrient category in which broccoli's antioxidant benefits stand out. Concentrated in broccoli are flavonoids like kaempferol and quercetin. Also concentrated are the carotenoids lutein, zeaxanthin, and beta-carotene. All three of these carotenoids function as key antioxidants. In the case of lutein and beta-carotene, broccoli has been shown not only to provide significant amounts of these antioxidants but to significantly increase their blood levels when consumed in the amount of 2-3 cups per day.

Source http://news.bbc.co.uk/1/hi/health/7201918.stm