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- Math is food for the brain. It helps you think precisely, decisively, and creatively. And, it helps you look at the world from multiple perspectives.
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- A teacher wants to take a break so she asks the class to add up all the numbers to 100. Before she can get to her desk a student says 5,050.
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- How did you know that answer?” The other students had barely reached a sum of 14.
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- Well numbers 1 to 100 come in pairs, 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ….. 50 + 51 = 101. Do that 50 times and you get 50* 101 = 5,050.
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- ………….. ( actually the general formula is n( n+1) / 2 = 100 ( 101) / 2 = 5,050)
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- A little frustrated the teacher comes up with a new lesson plan. Show me how the sum of cubes of a series of numbers is always equal to the square of the sum of those numbers:
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------------------- Any series, but, let’s use 1,2,3
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----------------- 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36
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---------------- ( 1+2+3)^2 = 6^2 = 36
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- How does this always work for every series of positive numbers?
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- Silence!
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- Let’s say you walk into a casino with $60 and you say you will leave with $100 or go bust. What are your odds with 50-50 fair play? Well, the math is at the end of this review, but, it is easy with 50% odds, it’s 60 / 100 or 60%.
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- Casinos do not make money with 50% odds.
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- At the craps table the odds are 49.3% . Your chance of winning $100 is 28%.
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- At the roulette table with odds of 47.3% your chance of winning $100 is 1.3%.
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- If you are an expert at Blackjack and know when to hold them and when to fold them your odds could reach 51%. In that case your chances of walking out with $100 are 93%. If you must gamble learn to play Blackjack.
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- If you must gamble learn to do the math. The formula is at the end of this review:
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- If you square a number that ends in 5 you can do it in your head. The answer ends in 25, always. The first digits of the answer are the first digit of the original number times the next higher digit.
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-------------------- 35^2 = 3*4 = 12 , Answer = 1225
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--------------------- 55^2 = 5*6 = 30 Answer = 3025
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------------------ 65^2 = 6*7 = 42 Answer = 4225 Remember this one we will use it later.
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- Now let’s do the math for draw poker. The rule is the number of ways to pick “k” objects from “n” objects when the order is not important : Here is the formula:
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------------------- n! / k! ( n - k )!
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----------------- “factorial” , “!” is the series of multiplications 5! = 5*4*3*2*1 = 120
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- How many 5 card poker hands are possible in a 52 card deck?
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------------------ n! / k! ( n - k )! = 52! / 5! ( 47!) = 52*51*50*49*48 / 120
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- The 47! cancel out and the answer is 311,875,200 / 120 = 2,598,960
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- There are over 2 ½ million poker hand combinations you can draw from a 52 card deck, where order is not important.
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- What are your chances of drawing a” flush”?
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- There are 4 suits of 13 cards in 52 cards. So, there are 4 chances of getting 5, “ k”, same suits out of 13 , “n”, cards.
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--------- 4 * n! / k! ( n - k )! = 4 * 13! / 5! * 8! = 4 * 13*12*11*10*9 / 5 *4*3*2*1! = 617,760 / 120 = 5,148
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- So, there are 5,148 chances for a flush out of 2,598,960 possible hands. That is 0.2 % chance of getting a flush:
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--------- 5,128 / 2,598,960 = 0.00198 That is 0.2 %
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- What are the chances of drawing an ace? Since there are 4 aces. You could draw 1 ace, 2 aces, 3 aces, or 4 aces in 5 cards. To make this problem easier, do the opposite. What is the possibility of not drawing an ace, there are 48 of those chances.
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------------------- n! / k! ( n-k )! = 48! / 5! * 43! = 1,712,304
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-------------------- 2,598,960 - 1,712,304 = 886,656 which is 34%
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- You have a 34% chance of drawing an ace in a poker hand when it comes out of 52 cards.
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- Think of any number, double it, add 10, then divide by 2. Now subtract your any original number. What is the answer? ……………………… 5.
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----------------- It is always 5. why?
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---------------- 2*n + 10 / 2 = n + 5
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---------------- n + 5 - n = 5
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- On April 7, 2014 Delaney will be twice as old as James. Two years ago Delaney was 3 times as old as James. How old will Delaney be on her birthday?
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--------------------- Delaney = 2 * James
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-------------------- Delaney - 2 = 3 ( James - 2 )
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-------------------- 2 * James -2 = 3 James - 6
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--------------------- James = 4
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----------------- Delaney = 2* James = 8
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- Delaney will be 8 years old on April 7, 2014.
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- Remember when we squared a number ending in 5:
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---------------- 65^2 = 6*7 + 25 = 4, 225
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-------------- Where does this come from?
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--------------- (x + y) * (x - y) = x^2 - y^2
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- This is a special case since the products “+ xy” and “- xy” cancel. Normally, expanding this multiplication gives you x^2 +xy - xy + y^2. And, 5 is half way between 0 and 10, so the y^2 is always 25. See explanation at the end of this Review.
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- What is the square of 13?
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- Well think of it as ( 13 + 3 ) * (13- 3) + 3^2 = 160 + 9 = 169
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---------------- This one works because x^2 = x^2 - y^2 + y^2
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--------------- But we have already shown that x^2 - y^2 = (x + y) * (x - y)
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---------------- x^2 = (x + y) * (x - y) + y^2
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--------------- 13^2 = (13 + 3) * (13 - 3) + 3^2 = 16 * 10 + 9 = 169
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- What is the square of 99?
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--------------- 99^2 = ( 99 +1) * ( 99 - 1 ) + 1^2 = 100 * 98 + 1 = 9,801
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- It is not always easy to have a “+y” and a “-y” in your equation to arrive at easy numbers you can multiply in your head. So, you can make it more general with the formula: Try this several times, you need practice to do magic for your friends.:
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---------------- ( z + a ) * ( z + b ) = z^2 + za + zb + ab = z ( z + a + b ) + ab
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- To see how this works multiply 106 * 109:
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----------------- ( z + a ) * ( z + b ) = z ( z + a + b ) + ab
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---------------- ( 100 + 6 ) * ( 100 + 9 ) = 100 ( 115 ) + 54 = 11,500 + 54 = 11,544
---------------------------- multiply 107 * 111:
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---------------- ( 100 + 7 ) * ( 100 + 11 ) = 100 ( 118 ) + 77 = 11,800 + 77 = 11,877
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---------------------------- multiply 97 * 106:
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---------------- ( 100 -3 ) * ( 100 + 6 ) = 100 ( 103 ) - 18 = 10,300 -18 = 10,282
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---------------------------- multiply 14 * 17:
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---------------- ( 10 + 4 ) * ( 10 + 7 ) = 10 ( 21 ) + 28 = 210 + 28 = 238
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---------------------------- multiply 62 * 68:
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---------------- ( 60 + 2 ) * ( 60 + 8 ) = 60 ( 70 ) + 16 = 4200 + 16 = 4,216
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----------------------------Remember 65^2 = 65 * 65:
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---------------- ( 60 + 5 ) * ( 60 + 5 ) = 60 ( 70 ) + 25 = 4,225
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- Here is the probability for the gambling:
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-------------------------- “p” = probability with each bet that you win
-------------------------- “ (1-p)” = probability with each bet that you loose.
------------------------- “d” = dollars that you start with, $60.
------------------------- “n” = goal you reach to quit, $100
----------------------- Your probability of reaching your goal:
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---------------------- 1 - (( 1-p) / p) ^d / 1 - (( 1-p) / p) ^n
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---------------------- 1 - (( 0.49) / 0.51) ^60 / 1 - (( 0.49) / 0.51) ^100
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---------------------- 1 - (0.961) ^60 / 1 - (0.961) ^100
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---------------------- 1 - 0.091 / 1 - 0.187
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---------------------- 0.909 / 0.981 = 93%
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- If you can play expert Blackjack with 51% chance of winning you have a 93% chance of using $60 to make $100. Good Luck!
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- Pick any 4-digit number. Scramble those numbers into a different number. Subtract the smaller number from the larger number. Add up the digits of that number, if necessary add the next set of digits, eventually you will end up with the number 9. Trust me.
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- Other Reviews available upon request:
- 1650 Power of combinations
- 1638 Rent or buy
- 1565 Speed of satellites
- 1521 Calculating interest
- 1467 Calculating growth and decay
- 1329 Exponential curve
- 1459 Modular math
- 1344 Rocket launch
- 1284 String theory
- 1281 Many more math examples.
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RSVP, with comments, suggestions, corrections. Index of reviews available ---
--- Some reviews are at: -------------------- http://jdetrick.blogspot.com -----
---- email request for copies to: ------- jamesdetrick@comcast.net ---------
---- https://plus.google.com/u/0/ , “Jim Detrick” ----- www.facebook.com ---
---- www.twitter.com , --- 707-536-3272 ---- Thursday, March 26, 2015 ---
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