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Mrs. “anon for this”,
Here’s how it works,
Instead of x and x + 2, let’s use (x – 1) and (x + 1). Now FOIL (Firsts, Outers, Inners and Lasts) gives us (x – 1)*(x + 1) = x^2 + x – x – 1^2 = x^2 -1.
So 6 x 8 = (7 – 1)*(7 + 1) = 7^2 – 1 = 48.
This will also work for (x – 2)*(x + 2) = x^2 – 4 and so on.
Here’s a trick I figured out for squaring an integer where the units digit is 5. Let’s square 75. Take all the digits besides for the last one (in this case it’s just the number 7) multiply it by by that number + 1 (7 * 8 = 56) and attach the numbers 25 at the end (5625). Please check that 75 * 75 = 5625.
Proof (10x + 5)^2 = 100x^2 + 50x + 50x + 25 = 100x^2 + 100x + 25 = 100(x^2 + x) + 25 = 100 * x * (x + 1) + 25.
Now try it for 45 and move the decimal two spots to the left for 4.5 ^ 2. You should get 20.25.
Dr. Pepper,
Also, the math for x+0.5 works like this:
x*(x+1)=x^2+x
(x+0.5)^2=(x+0.5)*(x+0.5)
Dr. Pepper,
Excellento! Correct mathematicaly on both questions. Though the only place in the world with equal Fahrenheit and Celsius is Antarctica.
Dr. Pepper,
The math for x+0.5 works like this:
x*(x+1)=x^2+x
and
(x+0.5)^2=(x+0.5)*(x+0.5)=x^2+0.5x+0.5x+0.5*0.5=x^2+x+0.25
So the square is 0.25 larger than the other product.
I did figure out the math when my daughter told me the trick because she was younger then and used smaller numbers, so I wanted to prove it would work for all numbers.
Thanks for the proof for squares of numbers ending in 5. I’d heard of that but didn’t know why it worked.
Note to mod: Would you please delete my previous reply to Dr. Pepper? I hit “send post” too soon. Thanks.
anon for this
(x+0.5)*(x+0.5) = x^2 +.5x + .5x +.25 = x^2 + x +.25.
=> 4.5 * 4.5 = 4^2 + 4 + .25 = 20.25
I agree that this will work but I found that my students had a hard time performing more than two mathematical operations mentally. (x+0.5)*(x+0.5) = x^2 +.5x + .5x +.25 reduces to three terms x^2 + x +.25 , by (x – 2)*(x + 2), the two middle terms cancel out so there are only two terms left, x^2 – 4.
If you (or your daughter) have no issue with this consider yourself gifted.
In a house with all southern exposure – that would have to be at the North Pole, so the bear would be a polar bear.
Dr. Pepper,
My daughter didn’t figure out the proof; she was only 5 years old at the time. She just figured out the rule using examples like 3*5 and 4^2. I did the proof on paper once to satisfy myself that it would always work. Now I just use the shortcut to quickly solve math problems (like estimating the area of a 9″ diameter pie pan).
GivPerf – Right. The answer to the second one is the same (though the reasoning involves knowledge of physics). The answer “red” was given in my last discussion too. Insightful, but misses the point.
Joseph, have you never been to Yukon? Your statement is far from being correct. And I might add that the temperature at the South Pole can go South of -100 degrees.
just googled it. lowest recorded is -129F
I think we’re seeing a whole new side to Joseph based on the comments of this day and the last. Is it possible that his screen name has been usurped by an imposter?
Dr. Pepper, are you also an engineer? I might have missed what you do.
I love all the math questions! It brings me back some good memories of math for fun 🙂
Here’s another one from an upper level math course (that can be answered in English with basic mathematics skills):
Does there exist a prime number “P” such that there is no prime number greater than “P”? (Is there a highest prime number?)
If “P” exists what is it, if not why does “P” not exist?
SJSinNYC
My Ph.D. is not in engineering but I did take some engineering courses. If you have some questions I’ll be happy to try to help you.
anon for this-
Dr. Pepper-
The puzzle was, “if you have a square wall made up of square bricks and remove one column from the side and add one row to the top, what will the difference in brick-count always be?”
SJSinNYC-
Math ? fun!
Math <> fun!
Math^2 = fun! (a negative squared)
Dr. Pepper-
One question:
The numbers 3, 4 and 5 (used together) are very helpful for builders and contractors.
How?
squeak – i totally agree.
joseph – you could save yourself by saying that your wife or kid got on here today…
Flights of Fancy
Airplanes, ships, cars, darts and many other objects are smooth to ensure less air resistance and faster speeds.
Why then are golf balls dimpled?
I can only try
(X + 2 * Y = 98) minus (X + Y = 74) => Y = 24.
We are subtracting one equation from another. (Cramer’s rule would also work.)
3^2 + 4^2 = 5^2 (is that what you were thinking of?) there are an infinite more sets of three integers that will fit that relationship.
The dimples in Golf balls allow the balls to travel further by trapping air inside of them and allowing the ball to ride or “float” on the air. A more detailed explanation is beyond the scope of this thread.
ICOT:
Perhaps this is easier to follow…
HM = Human, HR = Horse
HM + HR = 74
2HM + 4HR = 196
(2HM + 4HR) – (2 HM + 2HR) = 196 – 148
2HR = 48
HR = 24
HM + (24) = 74
HM = 74 – 24
HM = 50
24 horses and 50 humans.
Dr. Pepper-
a) I got it – thanks a lot!
b) You’re headed in the right direction, but what does that have to do with builders, contractors and handymen? Also – I thought I remembered learning once that there are no other integers that when squared add up to the square of a third integer.
c) Flights of Fancy – Correct! (Don’t forget that as well as adding lift, it also decreases drag by carrying air in the dimples to the “rear” of the ball. Agreed – the full technical explanation is not for here).
Joseph-
Thank you for your explanation, too.
<<Math + me = Rebbe Akiva’s water + rock>>
ICOT: Put look what happened to that rock… and look where it got Rebbe Akiva!
Every month, a girl gets allowance. Assume last year she had no money, and kept it up to now. Then she spends 1/2 of her money on clothes, then 1/3 of the remaining money on games, and then 1/4 of the remaining money on toys. After she bought all of that, she had $7777 left. Assuming she only gets money by allowance, how much money does she earn every month?
Joseph-
Working our way backwards:
7777 x 1 1/3 = 10369.33
10369.33 x 1.5 = 15554
15554 x 2 = 31108
“Assuming she only gets money by allowance, how much money does she earn every month?”
More than I do, apparently…
I can only try
There are actually an infinite amount of integers that will satisfy the equation a^2 + b^2 = c^2, another common example is 5, 12 & 13. There are no integers for a, b, c and n such that a^n + b^n = c^n for n > 2. (Please don’t ask me to prove Fermat’s Last Theorem here.)
Of the many uses of Pythagorean Triples, as they are called, the most common one is finding the length of the hypotenuse of a right triangle given the length of the legs. If the legs measure 3 and 4 respectively then the hypotenuse will measure 5.
Joseph
15,554,
7777 * 4 * 3 * 2 =186,648
186,648 / 12 = 15,554.
ICOT,
3,4, and 5 are a pythagorean triple. That is, they can form the sides of a right triangle, where 5 is the side opposite the right angle. Then, according to the Pythagorean Theorem, 3^2 + 4^2 = 5^2 (9+16=25). Another example of a unique Pythagorean Triple would be 5,12, and 13.
ICOT, correct. And divide the annual salary you quoted, 31,108, by 12 for the monthly allowance of 2592.33.
Dr. Pepper, you took another route to get there; I believe the final step is missing.
Dr. Pepper-
anon for this-
What I should have said is that there are no sequential numbers where a^2 + b^2 = c^2 except 3, 4 and 5.
I read in the news several years ago that someone claimed he had discovered a proof for Fermat’s Last Theorem, but a mistake was uncovered. Then, a couple of years ago, someone else claimed he had discovered it, and mathematicians were very excited and preliminary indications were that he was correct, but it would take months to review and verify. I don’t remember if it was actually verified or disproven. What was the end of that story?
Builders and contractors often use the 3, 4, 5 measurement to ensure something is being built correctly at right angles (measure the sides at 3′ and 4′ from a corner – a straight line between those two points should be 5′). For all newlyweds building bookcases, this is an easy way to confirm that your parallelogram is also a rectangle 🙂
Joseph-
I didn’t divide the final result by 12, because I was misupek if we should count from Tishrei or January (just kidding – I simply forgot).
For (and from) Kids
Kid: Daddy, can you jump higher than a house?
Father (chuckling): No, of course not.
Kid: Daddy, do you have a green key?
Father: No.
Kid: A red key?
Father: No.
Kid: A blue key?
Father: No.
Kid: A pink key?
Father: No.
Kid: Yes you do. You have one on each hand!
Kid: Hot. You can’t catch a hot!
ICOT & Dr. Pepper,
I can prove a^n+b^n=c^n is false for all n>2. Oops, just ran out of space.
Joseph,
I read the question too fast too late at night. I thought it said that half was remaining, then one third was remaining, then one quarter was remaining…
I can only try
Andrew Wiles announced a proof in 1993 which was almost complete but a colleague of his, I think his last name is Katz, found a “hole” in the proof. Andrew Wiles spent about another year with one of his students “plugging the hole” and I far as I know the proof has been accepted.
This might be a legend but I heard that in the 1800s and early 1900s there were large rewards set aside for the first one to prove (or disprove) the theorem. Unfortunately for Wiles the rewards were in German Marks held in German banks and after the record inflation Germany suffered after World War I, the rewards were practically worthless.
Joseph,
Alright, here it is:
( [3x – 3x^2 +1]^744 ) x ( [- 3x + 3x^2 +1]^745 ) ?
Actually I’m not sure what you mean by the sum of the coefficient (btw, wouldn’t that be product?)
((3 – 9)^744)((-3 + 9)^745)
(-6)^744
x 6^745
_______
1
or do you mean
( [3x – 3x^2 +1]^744 ) x ( [- 3x + 3x^2 +1]^745 ) ?
((3x – 9x +1)^744)((-3x + 9x +1)^745)
((-6x +1)^744)((6x+1)^745)
Well actually I guess I get one in any case. or is that 2? I really don’t remember.
or is it 6x+1?
(copied and pasted the post from the other thread.)
What’s the answer?
Dr. Pepper – thanks for the engineering help/offer, but I’m already an engineer! Anon is too (if I’m not mistaken).
SJS, yes, but I don’t work outside the home now.
Can someone be kind enough to respond to intellegent? She wants the sum of the coefficient of the equation she put in the beginning of her post (3rd line. I’m away from a computer now.)
Anon, it doesnt mean you arent still an engineer 🙂
SJS, you’re right, I definitely still think like one.
Joseph:
Assuming she only gets money by allowance, how much money does she earn every month?
I disagree and say you are all wrong, including Joseph. The correct solution to this riddle is $0. It doesn’t say stipend.
Anon, given your aptitude for math you can surely tell 🙂
Plus, as a friend of my mother’s used to say you are now a “domestic engineer.” 🙂
noitallmr-
Since the father who played the straight-man was yours truly, they were riddles from my perspective.
Here’s a tough one from a magazine back in the ’50’s. If anyone here can solve it without using brute force I will be very, very impressed. I might even bestow upon you all of my designations:
5 men are shipwrecked on a desert island. On the first day, they begin collecting food. The only food source on the island is coconut. The men spend the whole day collecting coconuts and at the end of the day have quite a nice pile. Exhausted, they all lie down to sleep.
After the others have fallen asleep, one man realizes that any of the others could steal from his “share” while the group is sleeping. So he gets up and divides the coconuts into 5 even piles. But there is one extra coconut – so he gives it to a monkey who is jumping around in the nearby tree. The man takes his pile and hides it, moves the other 4 piles back together to hide what he did, and then goes to sleep.
A short while later, one of the other men wakes up, and reasons the same as the first man, that he had better take his share now so that he does not get cheated. He divides the big pile into 5 even ones, but there is one left over. He gives it to the aforementioned monkey. He then takes his pile, pushes the other 4 piles back into 1, hides his and goes back to sleep.
A short while later, another man wakes up and does the same as the man before him. Again there is an extra coconut when he divides by 5, which he gives to the monkey. This process continues until every man has done the same deed. That said, each man now has a hidden pile of coconuts, there is a diminished pile left for the community, and the monkey has 5 coconuts.
In the morning, they all wake up and divide the pile into 5. This time, it divides evenly and the monkey walks away dejected. Each man must have noticed that the pile was much smaller than last night, but his guilt prevented him from bringing up the subject.
How many coconuts were there originally, and how many were there when it was finally divided?
There are of course mulitple solutions to this (I found over 100), but they are all multiples of the minimum solution. So there is really only one solution of importance – the minimum. Using brute force requires plenty of skill in this problem so it is legitimate to answer it that way, but an elegant solution would be amazing!
squeak – she does the laundry to earn her allowance. Sorry for skipping that crucial detail. 🙂
Joseph – then it’s not a riddle – it’s just a math problem (9th grade, I would say)
squeak – then how’d 2 (very) kluger yidden not get a 9th grade Q.?
Are you saying that it is a riddle because a couple of people got the wrong answer? Can’t argue with logic like that….
squeak
This is not as innocent as it looks.
Let the original pile = y and the final pile (before it’s divided into 5) = x.
We know that x must be divisible by 4 and 5 (the last person split it into 4 even groups and the 5 people split it into 5 even groups).
y = ((((((x)*5/4+1)*5/4+1)*5/4+1)*5/4+1)*5/4+1) which simplifies to (3125*x)/1024 + (8404/1024).
(8404/1024) = 8 + 53/256 therefore the fractional part of (3125*x)/1024 must equal 203/256 (so we are not left with a fraction at the end).
=> (3125*x)/1024 = an integer (henceforth denoted as “I”) + 203/256
=> (3125*x)/1024 = I + 203/256
=> 3125*x = 1024*I + 812
So we need MOD(3125*x,1024) = 812
MOD(3125,1024) = 53
MOD(812,53) = 17
MOD(53,17) = 2
=> x = 4 * 5 * MOD(812,53) * (MOD(53,17) + 1) = 4 * 5 * 17 * 3 = 1020
=> x = 1020 and y = 3121
If we weren’t looking for the minimum value of x the correct answer would be 1020 + 1024c where c is a non-negative constant.
………………………………………..
<closes mouth>
Dr. P, please tell me that you’ve already seen this problem. I guess that would make you an old timer like me (with quite a memory), but that’s better than believing that you came up with that in 1.5 hours (and that’s if I assume you were working on it the whole time).
Now that we’ve been introduced, I’d like to ask a favor. Can you stop using the clam stuff please?
squeak-
Good puzzle.
The original number can end with either a 1 or a six (subtract 1, get a number divisible by 5)
the next four numbers must all end with 16, 36, 56, 76 or 96 (divisible by 4, subtract 1 divisibe by 5)
the last number must be 0 ot 5 (divisible by 5)
From that point on, brute force produces the following:
3121
2496
1996
1596
1276
1020
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