Posted tagged ‘time’

AUTOMOBILE ENGINES

September 10, 2011

The working of an automobile engine follows the same principle as an internal combustion engine. Air, from outside, enters the engine through the air cleaner and reaches the throttle plate.
The pedal in your car is the control for the amount of air that you would want to be taken in, and you control it by pressing on this gas pedal.
The air is then distributed through the intake manifold of the cylinders.

At some point fuel is injected into the air stream, and the mixture vaporizes and is drawn into the cylinders as they start their intake stroke.

This way, when the cylinder has reached its bottom, it has drawn in sufficient mixture. As it moves up, compressing the mixture, the spark plug ignites the mixture, and as the powerful gas formed expands, it pushes the cylinder to the bottom with the cylinder once again drawing in the mixture.

In designing automobile engines, you need to be a specialist in automobile engineering.
The consideration that is taken while designing such an engine is whether it should be a carburetor or a diesel one. carburetor engines are most commonly found in passenger cars and low capacity trucks, while trucks with a capacity over two tons are fitted with diesel engines, including dump trucks, trailer tractors and bus.

Increasingly the medium and low-capacity vehicles are being fitted with diesel engines, since the fuel consumption of these engines are 30% to 50% lower than the carburetor engines.
Diesel engines not only cost more, but maintenance is much more expensive than the other type of engine. Diesels require more metal parts per kilowatt.
The critical parts of diesel engines are made of alloy steel, and the fuel injection system is much more expensive than carburetor engines.

However, the cost of manufacturing carburetor engines has increased with the use of higher mechanical grade components, considering the thermal loads of the material used. At the same time the use of high alloys and increase in production costs have contributed to the higher price of such engines.

There is a sharp rise in using aluminum alloys in design of carburetor engines in passenger cars, and with the use of high octane petrol, the cost of operation of these cars have come down extensively. Using alloy steel in constructing the engine body and other parts of the engine, makes the car lighter and hence fuel consumption goes down substantially.

The main parts that are made of high steel alloy are the main casting of the engine, the cylinder head, water and oil pumps, oil filter housing, end covers of the generator and starter, and the intake pipes. It has been observed that by using high steel alloys, the weight of the car is reduced by 35%.

The power per liter, per unit of piston area, and the brake effective pressure are 6% to 8% lower in air-cooled engines, compared to engines having liquid cooling mechanism. This is due to the fact that in engines with liquid cooling there are great losses in cylinder charging caused by the high temperature in pipes, ducts in the head, cylinder walls and head, etc.

The size of air cooled engines are much bigger than the engines with liquid cooling having the same capacity, and this is because the cylinder axes difference is larger in air-cooled engines. Taking account of the radiator dimensions, if both engines are compared, the air-cooled engine will vary slightly with its height a little longer than or approximately the same length as the water-cooled engine. As far as the width and the height is concerned both engines are about the same.

The auxiliary units of the feed and ignition, and generator and starter systems are a bit difficult to fit on the body of the air-cooled engines, because of the presence of hoods and having a danger of over-heating.

Welding and Farming – The Two Go Hand-In-Hand

September 8, 2011


 

Welding and FarmingWelding and farming? They have more in common than you might think. In fact, one astute farmer recently noted, “you can’t run a farm without welding.” This farmer was absolutely correct — to keep equipment in working order for the critical seasons of planting and harvesting, welding and hardfacing during the off-season are musts. A good working knowledge of these processes also comes in handy when your equipment breaks down during off-hours and you need to quickly fix so you can continue your work.

In this article, we will introduce you to some of the key concepts in welding and hardfacing. When we refer to welding, we are talking about joining metal pieces together to build something. The weld is primarily for strength purposes. Hardfacing, on the other hand, is depositing (by welding with special hardfacing electrodes) wear- resistant surfaces on existing metal components which are under stress to extend their service life. Hardfacing is very commonly done to metal edges that scrape or crush other tough materials -like the blade on a road grader.

Welding and FarmingWe will discuss different applications, ways to identify metallurgy, basic welding procedures and safety. So often, the beginning or novice welder will not get the desired results and assume his welding machine or electrodes are not working properly. In many of these instances, though, the farmer did not take the necessary preparations before welding or has chosen the wrong process, parameters or consumables. In this article, we hope to educate you so that you will know what to use in a few applications and can get the best results. Realize that although a little welding knowledge could help you a lot, there is a lot to becoming a true welding expert, which would cover many books!

Welding Applications

Welding and FarmingFarmers constantly need to repair and modify machinery and equipment to suit their specific needs. This instant ability to alter steel gates, chutes, animal pens, and machinery is such a tremendous benefit to the farmer. Repairing a broken plow or combine in the field by welding it where it broke in minutes can literally save an entire crop. The needs of beef cattle can usually be taken care of with mild steel. Dairy cattle, and virtually their entire milk-handling system require stainless steel. Two similar appearing animals with very different welding needs. But both needing welding to succeed.

Hardfacing Applications

There are many different items that could potentially benefit from hardfacing on the farm. They can basically be put into three “wear” categories – abrasion, impact, and metal-to-metal. Abrasion is one of the most common wears you will see on a farm, in this category falls all earth engaging implements such as tractor buckets, blades, teeth, grain handling products and feed mixers. Under the impact heading you will find equipment used to pound and smash such as crusher hammers. Metal-to-metal refers to wear from steel parts rolling or sliding against each other. Metal-to-metal wear occurs on such items as crane wheels, pulleys, idlers on track-drives, gear teeth and shafts.

Although farmers use welding and hardfacing techniques to rebuild old, worn-out components, Lincoln recommends hardfacing many new components as well. By hardfacing something that is new, it may increase the overall life expectancy of that product.

Basic Metallurgy

Before you can weld or hardface, you first need to identify the parent metal. A good rule of thumb on the farm is that nothing is mild steel. Almost all implements are high strength steels (either high or low alloy) and many are higher carbon steels. But how do you tell the difference? There are a couple of tests that can help.

Welding and FarmingThe first is a magnetic test. If a magnet will stick to the implement then it is likely iron-based. A magnet that will not stick indicates probably a manganese or stainless product. Secondly, try the spark test. If you take a grinder to the item, do you get 30″ long, moderately large volume of yellow sparks with just a few sprigs and/or forks indicating mild steel, or do you achieve 25″ long, slight to moderate volume of yellow orange sparks, a few forks with intermittent breaks but few if any sprigs to indicate alloy steels or do you get 15″ long short, red sparks in large volume with numerous and repeating sprigs, which are telltale signs of a high carbon metal? Another test, the chisel test, will help indicate the type of metal as well. If the metal fractures in large chunks when you take a chisel to it, this means you have cast iron, which can be very difficult to weld unless using special high-nickel electrodes and heat-treating. On the other hand, if the chisel yields corkscrew-like shavings, you are looking at a weldable steel.

What Is the Goal?

Now that you have identified the base material, you need to assess your final goal. In a farm type setting, you need to ascertain whether you need to strengthen the item or prevent wear? If the item in question is a hitch bar on a tractor, the ultimate goal is strength and ductility so that it will not break. WELD IT! If you are talking about an earth-engaging tool, you don’t want it to wear out. HARDFACE IT!

Identify What Method to Use

There are three types of welding methods to consider. They differ by speed and cost. The methods are all available to all welding and hardfacing products. However, specific products often have properties that are somewhat unique and not exactly duplicated when utilized by a different process.

Stick Welding

Manual or stick welding requires the least amount of equipment and provides maximum flexibility for welding in remote locations and in all positions. Typically, each rod permits welding for about one minute. In seconds, one can change from mild steel to stainless to hardfacing. In seconds, the electrode can change from small to large diameter for small or large welds. Although simplest, this type of welding takes the greatest operator skill.

Semiautomatic

This type of welding uses wire feeders and continuously fed electrodes. The welding gun is hand-held by the operator. The gun keeps feeding wire as long as the trigger is depressed. This is also much easier to learn than stick welding. This type of setup is becoming more popular on farms, which do more than minimal repair work. Semiautomatic welding increases deposition rates over manual welding because there is no need to stop after burning each rod.

Automatic

Requiring the greatest amount of initial setup, automatic welding has the highest deposition rates for maximum productivity. The welding gun is carried by a mechanized carriage and the welding operator just pushes a start button. This would rarely be found on a farm, but is common at repair centers for heavy equipment that would rebuild your parts for you if the schedule was mutually acceptable.

Welding Procedures

There are five basic steps when welding that must be followed.Welding and Farming

  • Proper Preparation – You first need to ensure that the metal you are welding is clean and dry. Remove rust, dirt, grease, oil and other contaminants by wire brushing. If not removed, these contaminants can cause porosity, cracking and poor weld deposit quality. You must also remove badly cracked, deformed or work-hardened surfaces by grinding, machining or carbon-arc gouging.
  • Proper Preheat – The combination of alloy content, carbon content, massive size and part rigidity creates a necessity to preheat in many welding or hardfacing operations. Most applications require preheating, as a minimum to bring the part to a room temperature of 70ƒ-100ƒ F. Medium to high carbon and low alloy steels may require higher preheat to prevent underbead cracking, welding cracking or stress failure of the part. Preheating can be done with either a torch, oven or electrical heating device. Special temperature-melting crayons can help you verify proper preheat. Too much heat and you can often ruin alloy materials!
  • Adequate Penetration – Correct Welding Procedure – Identify the correct amperage, travel speed, size of weld, polarity, etc. Make sure the completed weld meets your expectations in regards to size and appearance. Welds should be smooth and uniform, free from undercut or porosity. If possible, watch a video showing the type of welding you will be doing so you know what things are suppose to look like.
  • Proper Cool Down – Preheating is the most effective way of slowing the cooling rate of massive or restrained parts, which are inherently crack sensitive. Insulating the part immediately after welding with dry sand, lime, or a glass fiber blanket also helps minimize residual cooling stresses, weld cracking and distortion. Never quench a weld with ice or water as this will lead to greater internal stresses and potentially weld cracking.
  • Post Weld Heat Treatment – Some items may require tempering or heat-treating. What this means is that you warm the item up with your torch after welding and allow it to slowly cool.

Safety

There are a few rules you should follow as you are welding/hardfacing:
Welding and Farming

  • Protect yourself from fumes and gases – Always weld in an open, well-ventilated room and keep your head out of the fumes – especially with hardfacing
  • Wear protective clothing – Protect your eyes and face with a welding helmet designed for arc welding, not just gas welding goggles. In the same manner, protect your body from weld spatter and arc flash with woolen or cotton clothing, a flameproof apron and gloves, and boots. Also make sure to protect others around you from the arc rays as well.
  • Beware of electric shock – Do not touch live electrical parts and make sure that your welding machine is properly grounded. Never weld if you are wet or if your gloves have holes in them.
  • Fire/explosion hazard – Never weld in an enclosed space or near hay, feed bags, gasoline, diesel, hydraulic fluids or anything else that can be within the reach of your welding sparks that would cause a fire or explosion. Never weld alone. Always have a buddy nearby in case of an emergency.

Conclusion

After reading this article, you should be able to reap the benefits of welding in much the same way as you already reap the benefits of the earth on your farm.

Selecting Your Welding Process

Sure, you know you have a weld to make. . .that’s the easy part. . . but you need to start by examining your application.. Everybody’s job is individual and has specific requirements. Therefore, if you’re really confused the best idea is to consult a welding expert in person. If you still have questions after reading this article, just ask us online.

However, this article can help you with welding process selection in four easy steps:

1.) The joint to be welded is analyzed in terms of its requirements.

2.) The joint requirements are matched with the capabilities of available processes. One or more of the processes are selected for further examination.

3.) A checklist of variables is used to determine the ability of the selected processes(s) to meet the particular application.

4.) Finally, the proposed process or processes deemed most efficient are reviewed with an informed representative of the equipment manufacturer for verification of suitability and for more information

Step 1 – Analysis of Joint Requirements.

The first thing to look at is whether your weld joint is large or small, whether the joint is out-of-position or not, and whether the base metal is thick or thin.

In welding, the needs of any joint are expressed in four terms: Fast-Fill (high deposition rate), Fast-Freeze (the joint is out-of-position – overhead or vertical), Fast-Follow (high arc speed and very small welds), and Penetration (the depth the weld penetrates the base metal)

Fast-Fill is required when a large amount of weld metal is needed to fill the joint. A heavy weld bead can only be laid down in minimum arc time with a high deposition rate. However, Fast-Fill becomes a minor consideration when the weld is small.

Fast-Freeze implies that a joint is out-of-position, and therefore requires quick solidification of the molten crater. Not all semiautomatic processes can be used on fast-freeze joints.

Fast-Follow suggests that the molten metal follows the arc at rapid travel speed, giving continuous, well-shaped beads, without “skips” or islands. This trait is especially desirable on relatively small single-pass welds, such as those used in joining sheet metal.

Penetration varies with the joint. With some joints, penetration must be deep to provide adequate mixing of the weld and base metal and with others it must be limited to prevent burnthrough or cracking.

Any joint can be categorized in terms of the previously mentioned four factors. To determine the appropriate welding process, keep your efforts focused on the requirements of the weld joint. A joint that requires, or can be welded by, just one arc welding process is rare. In fact, the majority of joints usually are characterized
by a combination of these requirements to varying degrees. Once you’ve determined your appropriate joint requirements and ranked them, have your assessment reviewed by an experienced engineer or welder. With time and experience, you’ll be able to make these assessments more accurately and with less difficulty.

Step 2 – Matching Joint Requirements With Processes

Your equipment manufacturers’ literature usually will give information on the ability of various processes to fulfill the needs of the joint. (Or, a telephone call or email will bring the needed information.) A wrong answer is virtually impossible at this point, since the deposition rate and arc-speed characteristics of each process can be clearly defined. Since you have characterized your weld joint it is simply a matter of selecting the process that suits your characterization. To view some machines and consumables with various characteristics click here to view Lincoln Electric’s product line.

So what do you do when you find that two or more processes are suitable, which is sometimes the case? You create a checklist!

Step 3 – The Checklist

Considerations other than the joint itself have a bearing on selection decisions. Many of these are specific to your job or welding shop. However, they can be of great importance – and a key factor in eliminating alternate processes. Organize these factors into a checklist and consider them one-by-one:

Volume of Production. You must justify the cost of welding equipment by the amount of work, or productivity, required. Or, if the work volume for one application is not great enough, another application may be found to help offset the costs.

Weld Specifications. Rule out a process if it does not provide the weld properties specified by the code governing the work.

Operator Skill. Operators may develop skill with one process more rapidly than another. Will you have to train your operators in a new process? That adds cost!

Auxiliary Equipment. Every process has a recommended power source and other items of auxiliary equipment. If a process makes use of existing auxiliary equipment, the initial cost in changing to that process can be substantially reduced.

Accessory Equipment. Availability and cost of necessary accessory equipment – chipping hammers, deslagging tools, flux lay-down and pickup equipment, exhaust systems, et cetera – should be taken into account.

Base-Metal Conditions. Rust, oil, fit-up of the joint, weldability of the steel, and other conditions must be considered. These factors could limit the usefulness of a particular process.

Arc Visibility. Is there a problem following irregular seams? Then open-arc processes are advantageous. On the other hand, if there’s no difficulty in correct placement of the weld bead, there are “operator-comfort” benefits with the submerged-arc process; no head-shield required and heat from the arc is reduced.

Fixturing Requirements. A change to a semiautomatic process requires some fixturing if productivity is to be realized. Appraise the equipment to find out if it can adapt to processes.

Production Bottlenecks. If the process reduces unit fabrication cost, but creates a production bottleneck, its value is lost. Highly complicated equipment that requires frequent servicing by skilled technicians may slow up your actual production thereby diminishing its value.

The completed checklist should contain every factor known to affect the economics of the operation. Some may be specific to the weld job or weld shop. Other items might include:

  • Protection Requirements
  • Range of Weld Sizes
  • Application Flexibility
  • Seam Length
  • Setup Time Requirements
  • Initial Equipment Cost
  • Cleanliness Requirements

Evaluate these items realistically recognizing the peculiarities of the application as well as those of the process, and the equipment.

Human prejudice should not enter the selection process; otherwise objectivity is lost – when all other things are equal, the guiding criterion should be overall cost.

Step 4 – Review of the Application by Manufacturer’s Representative.

This may seem redundant, but the talents of experts should be utilized. Thus, the checklist to be used is tailored by the user to his individual situation. You know your application best and your welding expert knows his equipment best. Together, you should be able to confirm or modify the checklist. To contact a Lincoln Electric welding Expert click here.

Systemizing the Systematic Approach.

A system is of no value unless it is used. Create a chart and follow the steps to determining process. By taking the time to analyze each new weld joint, your operation will become more productive and your welding experience will be more fulfilling.

Source: Adapted from The Procedure Handbook of Arc Welding. The Lincoln Electric Company, 1994.

To order a copy of Lincoln Electric’s Procedure Handbook of Arc Welding or other welding textbooks and educational aids, click here to print out and fax an order form.

Arc-Welding Fundamentals
The Lincoln Electric Company, 1994.

Arc welding is one of several fusion processes for joining metals. By applying intense heat, metal at the joint between two parts is melted and caused to intermix – directly, or more commonly, with an intermediate molten filler metal. Upon cooling and solidification, a metallurgical bond is created. Since the joining is an intermixture of metals, the final weldment potentially has the same strength properties as the metal of the parts. This is in sharp contrast to non-fusion processes of joining (i.e. soldering, brazing etc.) in which the mechanical and physical properties of the base materials cannot be duplicated at the joint.

Fig. 1 The basic arc-welding circuit

In arc welding, the intense heat needed to melt metal is produced by an electric arc. The arc is formed between the actual work and an electrode (stick or wire) that is manually or mechanically guided along the joint. The electrode can either be a rod with the purpose of simply carrying the current between the tip and the work. Or, it may be a specially prepared rod or wire that not only conducts the current but also melts and supplies filler metal to the joint. Most welding in the manufacture of steel products uses the second type of electrode.

Basic Welding Circuit

The basic arc-welding circuit is illustrated in Fig. 1. An AC or DC power source, fitted with whatever controls may be needed, is connected by a work cable to the workpiece and by a “hot” cable to an electrode holder of some type, which makes an electrical contact with the welding electrode.

An arc is created across the gap when the energized circuit and the electrode tip touches the workpiece and is withdrawn, yet still with in close contact.

The arc produces a temperature of about 6500ºF at the tip. This heat melts both the base metal and the electrode, producing a pool of molten metal sometimes called a “crater.” The crater solidifies behind the electrode as it is moved along the joint. The result is a fusion bond.

Arc Shielding

However, joining metals requires more than moving an electrode along a joint. Metals at high temperatures tend to react chemically with elements in the air – oxygen and nitrogen. When metal in the molten pool comes into contact with air, oxides and nitrides form which destroy the strength and toughness of the weld joint. Therefore, many arc-welding processes provide some means of covering the arc and the molten pool with a protective shield of gas, vapor, or slag. This is called arc shielding. This shielding prevents or minimizes contact of the molten metal with air. Shielding also may improve the weld. An example is a granular flux, which actually adds deoxidizers to the weld.

Fig. 2 This shows how the coating on a coated (stick) electrode provides a gaseous shield around the arc and a slag covering on the hot weld deposit.

Figure 2 illustrates the shielding of the welding arc and molten pool with a Stick electrode. The extruded covering on the filler metal rod, provides a shielding gas at the point of contact while the slag protects the fresh weld from the air.

The arc itself is a very complex phenomenon. In-depth understanding of the physics of the arc is of little value to the welder, but some knowledge of its general characteristics can be useful.

Nature of the Arc

An arc is an electric current flowing between two electrodes through an ionized column of gas. A negatively charged cathode and a positively charged anode create the intense heat of the welding arc. Negative and positive ions are bounced off of each other in the plasma column at an accelerated rate.

In welding, the arc not only provides the heat needed to melt the electrode and the base metal, but under certain conditions must also supply the means to transport the molten metal from the tip of the electrode to the work. Several mechanisms for metal transfer exist. Two (of many) examples include:

  1. Surface Tension Transfer – a drop of molten metal touches the molten metal pool and is drawn into it by surface tension.
  2. Spray Arc – the drop is ejected from the molten metal at the electrode tip by an electric pinch propelling it to the molten pool. (great for overhead welding!)

If an electrode is consumable, the tip melts under the heat of the arc and molten droplets are detached and transported to the work through the arc column. Any arc welding system in which the electrode is melted off to become part of the weld is described as metal-arc. In carbon or tungsten (TIG) welding there are no molten droplets to be forced across the gap and onto the work. Filler metal is melted into the joint from a separate rod or wire.

More of the heat developed by the arc is transferred to the weld pool with consumable electrodes. This produces higher thermal efficiencies and narrower heat-affected zones.

Since there must be an ionized path to conduct electricity across a gap, the mere switching on of the welding current with an electrically cold electrode posed over it will not start the arc. The arc must be ignited. This is caused by either supplying an initial voltage high enough to cause a discharge or by touching the electrode to the work and then withdrawing it as the contact area becomes heated.

Arc welding may be done with direct current (DC) with the electrode either positive or negative or alternating current (AC). The choice of current and polarity depends on the process, the type of electrode, the arc atmosphere, and the metal being welded.

ADVANCE BATTERY STORAGE

August 26, 2011

01-EESTOR-Barium titanate Batteries-advanced battery storing technology-Ultra capacitor technology

For decades, battery storage technology has been a heavy weight on the back of scientific innovation. From cell phones to electric vehicles, our technological capabilities always seem to be several steps ahead of our ability to power them. Several promising new technologies are currently under development to help power the 21st century, but one small start-up looks especially well positioned to transform the way we think about energy storage.

01-barium_titanate_semi conductor-BaTiO3-Advanced Battery technology



Texas-based EEStor, Inc. is not exactly proposing a new battery, since no chemicals are used in its design. The technology is based on the idea of a solid state ultra capacitor, but cannot be accurately described in these terms either. Ultra capacitors have an advantage over electrochemical batteries (i.e. lithium-ion technology) in that they can absorb and release a charge virtually instantaneously while undergoing virtually no deterioration. Batteries trump ultra capacitors in their ability to store much larger amounts of energy at a given time.

EEStor’s take on the ultra capacitor — called the Electrical Energy Storage Unit, or EESU — combines the best of both worlds. The advance is based on a barium-titanate insulator claimed to increase the specific energy of the unit far beyond that achievable with today’s ultra capacitor technology. It is claimed that this new advance allows for a specific energy of about 280 watts per kilogram — more than double that of the most advanced lithium-ion technology and a whopping ten times that of lead-acid batteries. This could translate into an electric vehicle capable of traveling up to 500 miles on a five minute charge, compared with current battery technology which offers an average 50-100 mile range on an overnight charge. As if that weren’t enough, the company claims they will be able to mass-produce the units at a fraction the cost of traditional batteries.

“It’s a paradigm shift,” said Ian Clifford of ZENN Motor Co., an early investor and exclusive rights-holder for use of the technology in electric cars. “The Achilles’ heel to the electric car industry has been energy storage. By all rights, this would make internal combustion engines unnecessary.”

But this small electric car company isn’t the only organization banking on the new technology. Lockheed-Martin, the world’s largest defense contractor, has also signed on with EEStor for use of the technology in military applications. Kleiner Perkins Caufield & Byers, a venture capital investment firm who counts Google and Amazon among their early-stage successes, has also invested heavily in the company.

Sum it Up

August 24, 2011

Problem: you are given a sequence of numbers from 1 to n-1 with one of the numbers repeating only once. (example: 1 2 3 3 4 5). how can you find the repeating number? what if i give you the constraint that you can’t use a dynamic amount of memory (i.e. the amount of memory you use can’t be related to n)?
what if there are two repeating numbers (and the same memory constraint?)

Solution

as a programmer, my first answer to this problem would be make a bit vector of size n, and every time you see the number, set its correspond index bit to 1. if the bit is already set, then that’s the repeater. since there were no constraints in the question, this is an ok answer. its good because it makes sense if you draw it for someone, whether they are a programmer, mathemetician, or just your grandpa. its not the most efficient answer though.

now, if i add the constraint that you can only use a fixed amount of memory (i.e. not determined by n) and it must run in O(n) time… how do we solve it. adding all the numbers up from 1 to n-1 would give us a distinct sum. subtracting the total sum of all the numbers from the sum of n to n-1 ( which is (n)(n-1)/2 ) would give us the secret extra number.

what if you can only use a fixed amount of memory, and TWO of the numbers are repeated? we know that the numbers have a distinct sum, and the difference would be equal to the sum of our unknowns
c = a + b
where c is the sum and a and b are the unknowns – c is a constant
if we had another similar formula we could solve the two unknown equations. my first thought was that the numbers would have a distinct product – (n-1)!
if we divide the total product by the (n-1)! product, we would get another equation
c2 = ab
we could then solve the two equations to get them into quadratic formula notation
0 = ax^2 + bx + c
and solve for the two values of x. this answer is correct but factorial grows really fast.

some sort of sum would be better. the sum of the squares from n-1 to 1 would work. that would yield a function of the form
c2 = a^2 + b^2
which could also be solved by using the quadratic equation.

i think its fine to remind someone of the quadratic equation… (maybe only because i myself had to look it up to solve the problem) i mean really though, the last time i used it was probably in 10th grade. as long as they get the idea that given two unknowns and two equations you can solve for the unknowns – thats the point.

Daughters’ Ages

August 24, 2011

Two MIT math grads bump into each other at Fairway on the upper west side. They haven’t seen each other in over 20 years.

THE FIRST GRAD SAYS TO THE SECOND: “how have you been?”
SECOND: “great! i got married and i have three daughters now”
FIRST: “really? how old are they?”
SECOND: “well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there..”
FIRST: “right, ok.. oh wait.. hmm, i still don’t know”
SECOND: “oh sorry, the oldest one just started to play the piano”
FIRST: “wonderful! my oldest is the same age!”

problem: how old are the daughters?

Solution

solution: start with what you know. you know there are 3 daughters whose ages multiply to 72. let’s look at the possibilities…

AGES:            SUM OF AGES:
1 1 72            74
1 2 36            39
1 3 24            28
1 4 18            23
1 6 12            19
1 8 9             18
2 2 18            22
2 3 12            17
2 4 9             15
2 6 6             14
3 3 8             14
3 4 6             13

after looking at the building number the man still can’t figure out what their ages are (we’re assuming since he’s an MIT math grad, he can factor 72 and add up the sums), so the building number must be 14, since that is the only sum that has more than one possibility.

finally the man discovers that there is an oldest daughter. that rules out the “2 6 6” possibility since the two oldest would be twins. therefore, the daughters ages must be “3 3 8”.

(caveat: an astute reader pointed out that it IS possible for two siblings to have the same age but not be twins, for instance one is born in january, and the next is conceived right away and delivered in october. next october both siblings will be one year old. if a candidate points this out, extra credit points to him/her.)

this question is pretty neat, although there is certainly a bit of an ahafactor to it. the clues are given in such a way that you think you are missing information (the building number), but whats important isn’t the building number, but the fact that the first man thought that it was enough information, but actually wasn’t.

even if the candidate doesn’t know the solution, they could come up with some interesting thoughts. if they just stare at you and shrug “i dunno” then thank them for their time and don’t give them afogcreek pen.

Bumblebee

August 24, 2011

problem: two trains enter a tunnel 200 miles long (yeah, its a big tunnel) travelling at 100 mph at the same time from opposite directions. as soon as they enter the tunnel a supersonic bee flying at 1000 mph starts from one train and heads toward the other one. as soon as it reaches the other one it turns around and heads back toward the first, going back and forth between the trains until the trains collide in a fiery explosion in the middle of the tunnel (the bee survives). how far did the bee travel?

Solution

solution: this puzzle falls pretty high on my aha scale. my first inclination when i heard it was to think “ok, so i just need to sum up the distances that the bee travels…” but then you quickly realize that its a difficult (not impossible) summation which the interviewer could hardly expect you to answer (unless i guess if you are looking for a job as a quant). “there must be a trick” you say. eh, sort of i guess, enough to say that this question is a stupid interview question.

the tunnel is 200 miles long. the trains meet in the middle travelling at 100 mph, so it takes them an hour to reach the middle. the bee is travelling 1000 mph for an hour (since its flying the whole time the trains are racing toward one another) – so basically the bee goes 1000 miles.

there is no process to explain, so this question can’t possibly teach you anything about the person. they either know it or they don’t and if they already knew it before you asked, you’re not going to be able to tell when they give you the answer. so don’t ask this question. and if someone asks you this question, just tell them you’ve already heard it before.

100 Doors in a Row

August 24, 2011

Problem: you have 100 doors in a row that are all initially closed. you make 100 passes by the doors starting with the first door every time. the first time through you visit every door and toggle the door (if the door is closed, you open it, if its open, you close it). the second time you only visit every 2nd door (door #2, #4, #6). the third time, every 3rd door (door #3, #6, #9), etc, until you only visit the 100th door.

question: what state are the doors in after the last pass? which are open which are closed?

Solution

For example, after the first pass every door is open. on the second pass you only visit the even doors (2,4,6,8…) so now the even doors are closed and the odd ones are opened. the third time through you will close door 3 (opened from the first pass), open door 6 (closed from the second pass), etc..

question: what state are the doors in after the last pass? which are open which are closed?

solution: you can figure out that for any given door, say door #42, you will visit it for every divisor it has. so 42 has 1 & 42, 2 & 21, 3 & 14, 6 & 7. so on pass 1 i will open the door, pass 2 i will close it, pass 3 open, pass 6 close, pass 7 open, pass 14 close, pass 21 open, pass 42 close. for every pair of divisors the door will just end up back in its initial state. so you might think that every door will end up closed? well what about door #9. 9 has the divisors 1 & 9, 3 & 3. but 3 is repeated because 9 is a perfect square, so you will only visit door #9, on pass 1, 3, and 9… leaving it open at the end. only perfect square doors will be open at the end.