101. CHOOSE YEAR OR A DAY:
You awake inside a small transparent capsule sitting on the surface of Venus. From a small speaker you hear a voice that says, "We will leave you here either for a day or a year. If you choose to stay a day, we will give you $1 million. If you choose to stay a year, we will give you $2 million. Either way, you will have sufficient food and water. We will make sure the temperature is a constant 70 degrees Fahrenheit. We will also supply cable TV."
What is your choice? (Don't let money decide your answer).
Answer:
Choose to stay one year and win $2 million. Venus takes 243 Earth days to rotate on its axis, but it takes 225 Earth days to go around the sun. On Venus a day is longer than a year.
102. SIMPLE NUMBER THEORY:
Show that 1 + 1/2 + 1/3 + ... + 1/n is not an integer for n > 1.
Solution
Let 2m be the highest power of 2 ≤ n. Then none of the other positive integers ≤ n are divisible by 2m. Let k = lcm(1, 2, ... , n). Now write each of the terms 1, 1/2, ... , 1/n as fractions with denominator k. All will have even numerators except 1/2m which will have an odd denominator. Thus their sum is a fraction h/k with h odd, so it cannot be an integer.
Example:
1+1/2+1/3+1/4+1/5. Here n=5. 2m=22. Lcm=60 NOw.60/60 30/60 20/60 15/60 12/60. only in 1/22 did odd numerator comes.
103. PERCENTAGE RIDDLE:
'A' says to 'B', "You have 25% more money than I." 'B' replies to 'A',"You have 20% fewer money than I." Both are telling the truth. How is it possible?
Solution
In general the statements are true if 'A' has 4x money and 'B' has 5x money, where 'x' is any whole number.
104. BOUNCING DISTANCE:
A ball bounces up to a height exactly half the height from which it falls. If you drop it from a height of one metre, how far will it travel in toto before coming to rest?
Solution
After the first bounce, the ball rises to a height of half a metre and then falls the same distance. Thus it travels one metre between the first and second bounces. It then bounces up a quarter metre amd falls just as far, totaling half a metre between the second and third bounces. So, counting the height from which the ball fell initially (that is, one metre), the total distance travelled will be:
1+(1+1/2+1/4+1/8+...), i.e. three metres.
105. LONGEST NON-RE-TRACING DISTANCE:
"A small fly starts walking from one corner of a sugar cube(sugar candy) and confining herself to its edges, which are all 1 cm long, how far can it go without retracing any part of her path?"
Solution
The fly can utmost walk nine centimetres. Any different path covers 9 cm or less.
106. OPEN REFRIGERATOR:
If the doors of the refrigerator is kept opened, will it reduce the room temperature? or the opposite happens?
Solution
The room would heat up. When the refrigerator is open, heat from the room enters it and the motor runs in an attempt to maintain the low temperature of the inside. The device works by removing heat from the icebox and expelling it into the room and in order to do this, it has to consume electrical power. This also ends up as heat, which is expelled into the room. In other words, the refrigerator expels more heat into the room than it removes from the icebox, the discrepancy being due to the electrical power being consumed. The room, threfore heats up.
107. OVERFLOW OR NOT?
A certain inquisive boy took two ice cubes and put them into a glass. Then he filled the glass to the brim with water, ice cubes floating on the surface. Now the question is will the water overflow as the ice cubes melt?
Solution
The level of water would not be altered. The water's buoyancy in the glass supports the weight of the ice cube so that it floats. Since the ice cube weighs the same as the water it is composed of, if the ice cube were to melt,it's water would take up the same amount of space as the ice cube displaces. Thus, the water from the melted ice cube would fill up the space formerly occupied by the ice cube. Ergo, there will be no change of level in water.
108. TIME IN STILL WATER.
A fitness fanatic can swim at twice the speed of the prevailing tide. He swims out to a buoy and back again, taking 4 minutes in all. How long would it take him in still water?
Solution
It would take him 3 minutes to swim there and back in still water. Swimming against the tide, his speed is equal to that of the tide(though in the opposite direction). In still water it is twice the speed of the tide. Swimming with the tide, it is three times the speed of the tide. If we call the time it takes to swim the distance between the shore and the buoy when swimming against the tide, 1 unit, then with the tide it will take 1/3 unit and in still water 1/2 unit. To swim there and back, therefore, takes 1 unit in still water, 4/3 unit when there is a tide(since it takes one unit for the part of the swim that is against the tide, and 1/3 unit for the part of the swim that is with the tide). This is 1/3 as long again as the time taken to do the round trip in still water. Since it takes 4 minutes when there is a tide, it must take 3 minutes when there is no tide.
109. COWS,HAY AND OIL-CAKE:
It takes one bale of hay and 7 oil-cakes to feed adequately three cows. To feed four such cows, 2 bales of hay would suffice with the same number of oil-cakes as before. If there were no hay, how many oil-cakes would be required to feed 6 cows?
Solution
When the number of cows is increased from 3 to 4, the fodder rises by one bale of hay. One bale of hay, or its equivalent in oil-cakes, is the allowance per cow. If 3 cows need one bale of hay and 7 cakes, and since one more cow is adequately provided for by the additional bale of hay, 7 oil cakes will suffice for 2 cows. Therefore 6 cows would require 21 oil cakes in the absence of hay.
110. EQUAL OR UNEQUAL AND WHY?
Take a circle and inscribe 4 interior-disjoint congruent circles A, B, C, D that are cyclically tangent (A is tangent to B is tangent to C is tangent to D is tangent to A), and so that each is tangent to the large circle. A, B, C, and D are colored white in the diagram, below. Then, in each of these four, inscribe four congruent tangent circles in exactly the same way, to get 16 small circles, colored red in the diagram, below. Finally, inscribe a small circle C17, colored yellow in the diagram, in the space around the center of the large circle and tangent to each of A, B, C, D. Which is larger, the yellow C17 or one of the sixteen red circles inside A, B, C, D?
Solution Let's draw this whole thing on an x-y coordinate system. First, the outer circle. It will be the unit circle, centered at (0,0). Next, we'll draw circle A. Each white circle occupies a 90� sector of the larger circle. For convenience, I'll draw it in the upper right sector. Let r be the radius of the smaller circle. Its center, then, is (r,r) because it touches the axes. The distance from the origin to the center (r,r) is r*sqrt(2). Because the smaller circle is tangent to the larger one, r*sqrt(2)+r=1.
r*sqrt(2)+r=1
r*(sqrt(2)+1)=1
r=1/(sqrt(2)+1)
r=(sqrt(2)-1)
Now the radius of the circles inscribed in A bears the same relationship to circle A as circle A bears to the unit circle. The ratio of their radii is the same. So the radius of the inscribed circles is r^2 =
(sqrt(2)-1)^2 =
3 - 2sqrt(2)
The radius of the small circle centered at the origin, C17, is 1 minus twice the radius of circle A, which is
1-2(sqrt(2)-1)
1-2sqrt(2)+2
3-2sqrt(2)
It's the same radius as the 16 red circles.
