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.r.qxd 5/4/06 1:44 PM Page 73New and Improved Products: Intermediate Multiplication 73If you have an audience you want to impress, you can say2279,000 out loud before you compute 29.But this will not workfor every problem.For instance, try squaring 636:672362 2636 403,200 36 404,496366004042 236 1,280 4 1,296432Now your brain is really working, right? The key here is torepeat 403,200 to yourself several times.Then square 36 to get1,296 in the usual way.The hard part comes in adding 1,296 to403,200.Do it one digit at a time, left to right, to arrive at youranswer of 404,496.Take my word that as you become morefamiliar with two-digit squares, these three-digit problems geteasier.2Here s an even tougher problem, 863 :90028638??The first problem is deciding what numbers to multiplytogether.Clearly one of the numbers will be 900, and the othernumber will be in the 800s.But what number? You can com-pute it two ways:1.The hard way: the difference between 863 and 900 is 37 (thecomplement of 63).Subtract 37 from 863 to arrive at 826.Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 7474 Secrets of Mental Math2.The easy way: double the number 63 to get 126, and take the lasttwo digits to give you 826.Here s why the easy way works.Because both numbers arethe same distance from 863, their sum must be twice 863, or1726.One of your numbers is 900, so the other must be 826.You then compute the problem like this:900372 2863 743,400 37 744,769378264032 237 1,360 3 1,369334If you find it impossible to remember 743,400 after squaring37, fear not.In a later chapter, you will learn a memory systemthat will make remembering such numbers much easier.Try your hand at squaring 359, the hardest problem yet:400412 2359 127,200 41 128,881413184212 241 1,680 1 1,681140To obtain 318, either subtract 41 (the complement of 59)from 359, or multiply 2 59 118 and use the last two digits.2Next multiply 400 318 127,200.Adding 41 , or 1,681,gives you 128,881.Whew! They don t get much harder thanthat! If you got it right the first time, take a bow!Let s finish this section with a big problem that is easy to2do, 987 :Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 75New and Improved Products: Intermediate Multiplication 75What s Behind Door Number 1?he mathematical chestnut of 1991 that got everyone hoppingTmad was an article in Parade magazine by Marilyn vos Savant, thewoman listed by the Guinness Book of World Records as having theworld s highest IQ.The paradox has come to be known as the MontyHall problem, and it goes like this.You are a contestant on Let s Make a Deal.Monty Hall allows youto pick one of three doors; behind one of these doors is the big prize,behind the other two are goats.You pick Door Number 2.But beforeMonty reveals the prize of your choice, he shows you what you didn tpick behind Door Number 3.It s a goat.Now, in his tantalizing way,Monty gives you another choice.Do you want to stick with DoorNumber 2, or do you want to risk a chance to see what s behindDoor Number 1? What should you do? Assuming that Monty is onlygoing to reveal where the big prize is not, he will always open one ofthe consolation doors.This leaves two doors, one with the big prizeand the other with another consolation.The odds are now 50-50 foryour choice, right?Wrong! The odds that you chose correctly the first time remain1 in 3.The probability that the big prize is behind the other doorincreases to 2 in 3 because the probability must add to 1.Thus, by switching doors, you double the odds of winning! (Theproblem assumes that Monty will always give a player the option toswitch, that he will always reveal a nonwinning door, and that whenyour first pick is correct he will choose a nonwinning door at ran-dom.) Think of playing the game with ten doors and after your pickhe reveals eight other nonwinning doors.Here, your instincts wouldprobably tell you to switch.People confuse this problem for a variant:if Monty Hall does not know where the grand prize is, and revealsDoor Number 3, which happens to contain a goat (though it mighthave contained the prize), then Door Number 1 has a 50 percentchance of being correct.This result is so counterintuitive that Marilynvos Savant received piles of letters, many from scientists and evenmathematicians, telling her she shouldn t write about math.Theywere all wrong.Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 7676 Secrets of Mental Math1,000132 2987 974,000 13 974,169139741632 213 160 3 169310EXERCISE: THREE-DIGIT SQUARES2 2 2 21.2.3.4.409 805 217 8962 2 2 25.6.7.8.345 346 276 6822 2 29.10.11.431 781 975CUBINGWe end this chapter with a new method for cubing two-digitnumbers.(Recall that the cube of a number is that number mul-3tiplied by itself twice.For example, 5 cubed denoted 5  isequal to 5 5 5 125.) As you will see, this is not muchharder than multiplying two-digit numbers.The method is basedon the algebraic observation that3 2A (A d)A(A d) d Awhere d is any number.Just like with squaring two-digit num-bers, I choose d to be the distance to the nearest multiple of ten.For example, when squaring 13, we let d 3, resulting in:3 213 (10 13 16) (3 13)Since 13 16 13 4 4 52 4 208, and 9 13117, we haveBenj_0307338401_4p_c03_r1.r [ Pobierz całość w formacie PDF ]