Happy numbers The number 4599 is said to be 'happy', for on finding the sum of the squares of its digits, and then the sum of the squares of the digits of that sum, ans so on, the process ends in 1.
Numbers which do not end in 1 after this process are said to be 'sad'. Investigate! Which numbers are happy? What happens to sad numbers?
Twenty numbers in the range from 1 to 100 are happy, and they may be presented in a tree diagram showing the chain of numberswhich takes them to 1.
Know the time It takes only a few seconds' reflection to appreciate that once every hour the minute hand and the hour hand must point in precisely opposite directions. But how often in a day will each hand be pointing exactly at a minute division at the same time as the hands are precisely opposite each other?
When the surface of a lake or pond is undisturbed, it behaves like a plane horizontal mirror. The laws of reflection of light (angle of incidence equals angle of reflection) operate and only the light (from a point source on the opposite bank) reflected from a particular point of the surface can enter our eyes. This ensure that we see a clear image of the light source. However, when the surface becomes wavy due to the action of the wind, there are multiple points on it that are so inclined relative to us that they canall reflect the light into our eyes, and we see multiple images. As the waves move, these points also change and the images keep shifting.
Tractors and buffaloes A heavy crawler tractor is able to operate on soft, muddy ground but the farmer's as well as his buffaloes' feet sink. Why?
The answer lies in the difference between weight and pressure. Although the tractor is much heavier than the farmer or his buffaloes, its weight is distributed over a much larger area of its bottom surface. Consequently, the load carried by each square centimetre of its bottom surface (the "pressure") is fairly low. On the other hand, the weight of the farmer or his buffaloes is concentrated over the much smaller area of his feet or the hooves, producing a much higher "pressure". An object penetrates deeper not because it is heavier but because it exerts a higher pressure (force per unit area) on its support.
Singing Kettle Boiling water in a kettle is a daily chore for most of us. We are all familiar with the hissing sound (called the "singing" of the kettle) that starts a few after the kettle is put on the fire. This sound gradually increases and then suddenly drops when the water starts to boil. In fact, we know from the sudden drops of the sound that the water is ready, boiling. Have you ever wondered what causes the kettle to "sing"?
It is the bottom layer of the kettle gets heated first. As the temperature rises, steam bubbles (not air bubbles) from at the bottom. Being lighter than water, they rise and come in contact with the cooler layers of water above, contract, and eventually collapse. It is the collapse of a myriad steam bubbles that produces the hissing sound. The sound, therefore, increase as more and more steam bubbles from and collapse. Eventually, however, when the entire mass of water is heated the boiling point, the steam bubbles do not collapse ant more because they no longer encounter cooler layer of water. The hissing therefore ceases, and the whole mass of water in the kettle starts boiling.
Don’t Lick An Ice Tray Have you ever tried to hold a really cold frosted ice tray? If you have, you must have noticed your fingers tend to stick to the tray. Why? Don’t ever try to lick the tray - it will be a very painful experience!
There is always some moisture on your fingers. When you touch the frosted sides of the ice tray, this moisture freezes and the pressure of your fingers makes the frozen moisture stick to the ice crystals on the tray. If you try to lick the tray, your tongue will stick to it and a layer of the skin may be ripped off.
The Paper Kettle The picture shows an egg boiling in water in a paper pot. Impossible, you might think. Won’t the paper catch fire and the water spill over and put out the fire? Try the experiment yourself with a paper not made of some stiff paper and attach a piece of wire to it to enable you to hold it over the fire. The fire will lick the pot but nothing will happen to it, and you can boil an egg in it. What do you think is the reason?
The reason is that in an uncovered pot you can only heat water up to its boiling point, i.e., 100 deg.C. Water has a great capacity to absorb heat. It absorbs the heat that would have otherwise burnt the paper. In other words, if prevents the paper from being heated to a point where it would catch fire.
Tearing Wet Paper It is common experience that it is much easier to tear wet paper than dry paper. Have you ever wondered why?
It is the adhesive force between the cellulose fibres of which paper is made that must be overcome in tearing paper. In the presence of water this adhesive force which is of electrostatic origin is weakened, much the same way as when soluble like salt (e.g., sodium chloride) dissolve in water because of the weakening of the electrostatic attraction between the positively and negatively charged ions. In the case of paper the effect is perceptible because water wets paper and water molecules can flow into the spaces between the fibres, weakening the adhesive force between them.
Obedient Button A magician drops a shirt button into a glass filled with a carbonated beverage. It sinks to the bottom. A moment or two later, he waves his hand over the glass and says,"Button, rise!" The button floats slowly to the surface. When the magician snaps his fingers and says," Button, sink!" down it goes again. Can you tell the trick behind this magic?
This startling trick works automatically with any small light object. While it is resting on the bottom of the glass, tiny bubbles of carbon dioxide begin to cluster around it. When enough bubbles have collected to counteract the object's weight, they float it to the surface. On the surface, the bubbles burst and the weight of the object carries it down again. This up and down motion continues as long as there is carbonation in the liquid.
The Burning Flame Next time you carry a candle or a burning matchstick, notice that the flame is initially deflected backwards. Which way will it deflect if you carry it in a case or protect it with your hand?
Contrary to expectation, the protected flame will move forwards, not backwards! This is because the flame, being hotter is lighter than the surrounding air. Now, when a force is applied to a body, it moves faster, the smaller its mass. (This is Newton's second law of motion.) Being lighter, the flame moves faster than the surrounding air and is therefore seen to be deflected forwards.
Ice Fumes Have you noticed that when exposed to air, a large slab of ice appears to give out fumes? What are these fumes and why do they from?
When a large slab of ice is kept in the open, it gives out dense fumes. They are not fumes of any gas but simply water vapour that condenses in the cool air surrounding the ice. When the air surrounding the ice becomes very cold, some of the water vapour present in it condenses into tiny droplets of water. The condensed vapour looks like fumes when it moves up and down with convection currents of air.
Brim To Brim Stand two glasses completely filled with water, brim to brim inside a bowl, as shown. Move the upper glass a trifle to make a tiny, barely perceptible opening the aims, barely perceptible opening between the rims. Surface tension and the air pressure outside the glasses will prevent the water from escaping. Can you remove the water from the upper glass without touching either glass in any way?
The feat can be accomplished with a straw. Hold one end close to the opening the brims and blow through the other. Air will bubble up into the top glass, forcing water out through the opening and down into the bowl.
Centre The Cork Fill a glass with enough water to reach almost to the rim. Drop a small cork into the glass, and challenge anyone to make the cork float in the centre of the water without touching the sides of the glass. He will find it impossible. The cork always drifts one side.
After everyone has given up, show how easily it can be done. Add more water to the glass, pouring carefully from another glass until the water rises slightly above the rim. Because of surface tension, the water will form a convex surface, as shown. The cork naturally moves to the centre, where the water is highest, and there it will remain.
Can You Put Egg In A Bottle Can you drop an egg into a bottle and then get it out intact?
First remove the shell of the egg and then drop a burning match into the bottle just before you put the egg on the mouth. The flame uses up oxygen, thus creating a vacuum that draws the egg neatly into the bottle. Now to get it out intact, turn the bottle upside down so the egg falls into the neck. Tip back your head and blow vigorously into the inverted bottle as shown. When you remove your lips, the egg will pop out so quickly that you will be wise to keep your other hand near the opening so that you can catch it. This puzzle is a very good example to demonstrate the effect of air pressure.
Quadrupled! The four shapes shown, each of which can be thought of as consisting of half-squares, can be put together like the pieces of a jigsaw in four different ways to reproduce larger versions of themselves. Cut out the shaped from card and see if you can reproduce their enlargements.
The Floating Ball Cut a six-inch piece from the end of the straw. Put one end of the piece on your mouth, tip back your head, and hold a table-tennis ball a few inches above the other end. Blow as hard as you can, simultaneously releasing the ball. Instead of being blown away as you might expect, it remains suspended in mid-air. The harder you blow, the higher it floats above the straw. Can you tell the reason for it?
The explanation: When air is in rapid motion, its pressure is lowered. In this trick, the ball is actually imprisoned by the column of upward rushing air. As soon as it wobbles a bit to one side, the greater pressure outside the 'jet stream' forces the ball back into it again.
Unpepper The Salt This is an amusing dinner-table trick to show friends on dry winter days when static electricity is easy to produce. Shake a pile of salt on the tablecloth; flatten it with your finger, then shake some pepper on top of it. The problem is to remove the pepper from the salt,
Just put a static charge on a pocket comb by running it a few times through your hair. Bring one end of the comb to about an inch above the salt. The grains of pepper, which are lighter than salt grains, will jump to the comb.
The Magic Window Ask someone to jot down any three-digit number in which the first and last digits differ by at least 2. Suppose he writes 317. Tell him to reverse the digits and subtract the smaller number from the larger (713 minus 317 leaves 396). Finally, he must reverse the digits in this answer and add them to the answer (693 plus 396 equals 1089). 'Now if you will please breathe on that windowpane,' you say to him, pointing to one of the windows in the room, 'you'll see your final answer on the glass.' When he breathes on the glass, the number 1089 magically appear on the misted pale!
The secret is quite simple: the answer is always 1089. Before doing the trick, mix some detergent in a glass of water, dip your finger in the liquid, and write 1089 with the tip of your finger on the windowpane. The writing is invisible when dry, but when dry, but when someone breathes on the glass the area touched by your finger will not fog.
A healthy diet! Confirm this slogan by replacing each different letter by a different digit to form a correct sum.
An Easy Dissection Puzzle First, cut out a piece of paper or cardboard of the shape shown in the illustration. It will be seen at once that the proportions are simply those of a square attached to half of another similar square, divided diagonally. The puzzle is to cut it into four pieces all of precisely the same size and shape.
The solution to this puzzle is shown in the illustration. Divide the figure up into twelve equal triangles, and it is easy to discover the directions of the cuts, as indicated by the dark lines.
Two's Enough! Can you find a way of colouring in twelve of the small squares of the 6x6 board so that there are two coloured squares in each row and each column, and no more than two on any diagonal line?
Matchstick Machinations! In each of the arrangements of matchsticks, change the position of, but do not remove, four matches to make an arrangement of three squares.
Lift that Cube A single ice cube is floating in a glass of water. You hold a piece of string about four inches long. Problem: without touching the ice with your fingers, lift the cube out of the glass with the string.
Lay the string across the cube, as shown in the figure. Sprinkle salt on top of the ice. The ice around the string will start to melt. But in doing so, it will take in heat from the surrounding water which will refreeze about the string. After a minute or two, lift the string. The cube will adhere tightly to it!
Through the hole Place a halfpenny on a small square of paper, and trace a circle around it with a pencil. Cut out this circle as shown in the figure 1. Now can a penny be pushed through this hole without tearing the paper?
The surprising answer is yes. Fold the paper across the hole, with the penny inside Figure 2. It is now a simple matter to push the coin through the hole, as shown in Figure 3. For the trick to work it is only necessary for the circumference of the hole to be slightly more than twice the diameter if the coin to be passed through it.
Magic Dice Turn your back while someone tosses three dice. Ask him to: (1) add all the uppermost faces; (2) pick up one of the dice and add the bottom face to the former total; (3) roll this same dice again and add the number it shows on top to the previous total. Turn around and point out that you have absolutely no way of knowing which of the three dice was used for the second roll. Pick up the dice, shake them in your hand a moment, and then announce the correct total!
How do you know? Simple. Merely total the top faces of the three dice before you pick them up, and add seven. The fact used up here is that opposite sides of a dice always totals seven.
The two Aeroplanes A man recently bought two aeroplanes, but afterwards found that they would not answer the purpose for which he wanted them. So he sold them for $600 each, making a loss of 20% on one machine and a profit of 20% on the other. Did he make a profit on the whole transaction, or a loss? And how much?
The man must have paid $500 and $700 for the two machines, making together $1, 250; but as he sold them for only $1, 200, he lost $50 by the transaction.
The Cyclists' Tour Two cyclists were consulting a road map in preparation for a little tour together. The circles represent towns, and all the good roads are represented by lines. They are starting from the town with a star, and must complete their tour at E. But there arriving there they want to visit every other town once, and only once. That is the difficulty. Mr. Spicer said, 'I am certain we can find a way of doing it;' but Mr. Maggs replied, 'No way, I'm sure.' Now, which of them was correct? Take your pencil and see if you can find any way of doing it. Of course you must keep to the roads indicated.
When Mr. Maggs replied, "No way, I'm sure," he was not saying that the thing was impossible, but was really giving the actual route by which the problem can be solved. Starting from the star, if you visit the towns in the order, NO WAY, I'M SURE, you will visit every town once, and only once, and end at E. So both men were correct. This was the little joke of the puzzle which is not by any means difficult.
Dancing coin Can you make a coin to dance around the rim of an empty but still cold bottle?
When the bottle is empty but still cold, put a coin on the opening as shown in the figure. Dip your finger in water and let a drop or two fall around the edge of the coin to seal the opening. Now place both hands around the bottle, holding it firmly for about fifteen seconds. The coin will start to click up and down mysteriously. The fact behind this trick is the property of air that air expands when heated. The cold air inside the bottle is warmed by the heat from your palms. The expanding air escapes around the rim of the coin causing it to flutter.
Spawning Coins! Arrange twolve coins to form a square as shown. Now rearrange them to leave the square intact, but so that each side contains five coins instead of four.
Put 3 on 4, 6, on 7, 9 on 10, 12 on 1. Clearly this is not a unique solution; removing one coin from each side to put on a different corner would give the required result.
The Square Pack Given the identical square of card, how can you make one straight cut which will enable you to rearrange all the resulting pieces into one large square?
Put four of the squares on top of one another and cut through them from one corner to the middle of an opposite edge. Then arrange the pieces as shown to form the large square
Back-Packing! A mule and a donkey were walking along laden with sacks of corn. The mule said to the donkey, 'If you gave me one of your sacks, I would be carrying twice as much as you. But if I gave you one, we would both be carrying equal burdens.' How many sacks of corn were they each carrying?
The mule was carrying seven sacks of corn, and the donkey y sacks, then the mule's statements are equivalent to the equations: x + 1 = 2(y -1) x - 1= y + 1 These can easily be solved be solved simultaneously.
Romantic? From six you take nine And from nine you take ten Then from forty take fifty And six will remain.
Extrapolating from five seconds! (a) A clock strikes six in five seconds. How long does it take to strike twelve? (b) If five ladybirds devour five greenfly in five seconds, how many lady birds are required to devour a hundred greenly in a hundred seconds?
A Family Party A certain family party considered of 1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 4 children, 3 grandchildren, 1 brother, 2 sisters, 2 sons, 2 daughters, 1 father-in-law, 1 mother-in-law, and 1 daughter-in-law. Twenty-three people, you will say. No; there were only seven persons present. Can you show how this might be?
The party consisted of two little girls and a boy, their father and mother, and their father's father and mother.
Mamma's Age Tommy: 'How old are you, mamma?' Mamma: 'Let think me, Tommy. Well, our three ages add up to exactly seventy years.' Tommy: 'That's a lot, isn't it? And how old are you, papa?' Papa: 'Just six times as old as you, my son.' Tommy: 'Shall I ever be half as old as you, papa?' Papa: 'Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day.' Tommy: 'And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing-' Mamma: 'Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache.' Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma?
The age of Mamma must have been 29 years 2 months; that of Papa, 35 years; and that of the child, Tommy, 5 years10 months. Added together, these make seventy years. The father is six times the age of the son, and, after 23 years 4 months have elapsed, their united ages will amount to 140 years, and Tommy will be just half the age of his father.
Drawing A Spiral If you hold the page horizontally and give it a quick rotary motion while looking at the centre of the spiral, it will appear to revolve. Perhaps a good many readers are acquainted with this little optical illusion. But the puzzle is to show how I was able to draw this spiral with so much exactitude without using anything but a pair of compasses and the sheet of paper on which the diagram was made. How would you proceed in such circumstances?
Make a fold in the paper, as shown by the dotted line in the illustration. Then, taking any two points, as A and B, describe semicircles on the line alternately from the centres B and A, being careful to make the ends join, and the thing is done. Of course this is not a true spiral, but the puzzle was to produce the particular spiral that was shown, and that was drawn in this simple manner.
Find 6 numbers whose sum is 123456. Each of the numbers can only use the digit 1.
Can you make five 3's equal to 13? Here's how: 3 x 3 + 3 + 3/3 = 13 Now you try a few A. Make thirteen 1's equal to 13 B. Make thirteen 2's equal to 13 C. Make thirteen 9's equal to 13
Use digits 1 through 8 to fill the circles below. But wait! Make sure that no line connects consecutive numbers. That means 3 can't join 2 or 4; 6 can't join 5 or 7 and so on.
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