There are 8 marbles that weigh 1 ounce each, and 1 marble that weighs 1.1 ounces. The marbles are all uniform in size, appearance, and shape. You have a balance that contains 2 trays. You are only able to use the scale 2 times. How do you determine which marble is the heaviest using only the scale and marbles in 2 weighings. *A*: Place 3 marbles on each tray. - If the first weighing doesn't balance, remove all the marbles from the lighter side, and place one marble on each tray from the heavier tray. The heavier side is the 1.1 ounce marble, but if they balance, then the marble from the heavier tray from the first weighing that was not weighed in the second weighing is the heavier one(1.1). - If the marbles balance on the first wieiging, remove the marbles from the trays, and place 2 of the remaining unweighed marbles on the trays, one on each tray. If one is heavier, it is the heavier marble(1.1), but if they balance, the remaining unweighed marble is the heavier one ************************************ You are right next to a river and have a 5 gallon container and a 3 gallon container. You need to measure out 1 gallon of water. How do you do it? Answer: Fill up the 3 gallon container first. Pour it into the five gallon container. Fill up the 3 gallon container again, and fill up the rest of the 5 gallon container, and you should have 1 gallon left over in the 3 gallon container. ************************************ What number should replace the '?' to COMPLETE the following row of numbers? 6, 25, 64, 81, 32, ? *A*: The answer is 1. Why? Well you can write every number as a power of another number: 6 equals 6^1, 25 equals 5^2, 64 equals 4^3, 81 equals 3^4, 32 equals 2^5. Do you see what comes next? Well, first we took SIX to a power, then FIVE to a power, then FOUR to a power, and so on. So the '*' should be ONE to a power. But which power? Well, first we took a number to the FIRST power, then to the SECOND power, and so on. So the '*' should be to the power SIX. That makes that '*' should be ONE to the SIXTH power, which is 1x1x1x1x1x1 = 1. The following picture shows the formulas for 6, 25, 64, 81, 32 and (finally) 1. For those interested, you could lengthen the row to both sides. That would give: ..., 0.333, 0.125, 1, 6, 25, 64, 81, 32, 1, 0, 1, -512, ... ************************************ You have 10 bags with 10 marbles in each bag. All marbles weigh 10g except for the marbles in one bag which weigh 9g. But you don't know in which bag these 9g marbles are. You have to find out by taking zero or more marbles from zero or more bags and put them on a weightscale. After seeing the result of the weight, you should be able to tell in which bag the marbles weigh only 9g. So how many marbles from which bags do you take to weigh? *A*: Number the bags from 1 through 10. Take 1 marble from bag 1, take 2 marbles from bag 2, take 3 marbles from bag 3, ... take 9 marbles from bag 9 and take 10 marbles from bag 10. Now put them on the weightscale. Now there are 10 possible measurements. If all marbles would weigh 10g, the total would be 550g. But one or more marbles weigh 9g. So if you took one 9g marble, the total would be 549g. If you took two 9g marbles, the total would be 548g. And so on. You know the total weight, so you know the number of 9g marbles you took. That is, if the total was 549g, you took one 9g marble. If the total was 548g, you took two 9g marbles. And so on. Now that you know how many 9g marbles you took, you know in which bag they are because you took 1 marble from bag 1, 2 marbles from bag 2, etc. So if the weight is: 549g the bag with the 9g marbles is bag 1. 548g the bag with the 9g marbles is bag 2. ... 541g the bag with the 9g marbles is bag 9. 540g the bag with the 9g marbles is bag 10 *************************************** How do you get (exactly) 4 liter out of a lake if the only thing you have with you is a 3 liter and a 5 liter bowl? *A*: Fill up the 5 liter bowl (from the lake). Use the 5 liter bowl to fill up the 3 liter bowl. That leaves 2 liter in the 5 liter bowl. Empty the 3 liter bowl (back to the lake or trow it away, doesn't matter). Put the 2 liter from the 5 liter bowl into the 3 liter bowl. Fill up the 5 liter bowl (from the lake). Fill up the 3 liter bowl using the 5 liter bowl. The 3 liter bowl already contains 2 liter, so 1 liter goes from the 5 liter bowl to the 3 liter bowl. You just removed 1 liter from the 5 liter bowl. That leaves 4 liter and that's what we wanted!! **************************************** Try to make a total of 24 using exactly these 4 numbers: 3, 3, 7 and 7. You can use +, -, x, / and ( ). *A*: 7 * (3 + 3/7) The above equals (distribute 7 over +): = 7*3 + 7*3/7 And that equals: = 21 + 3 = 24 *****************************************