Tag Archives: numbers

A Nutty Problem

Mama Squirrel found a bag of nuts in the driveway of the house near her tree home. She decided to separate the nuts into equal portions for her children.When she tried putting 6 nuts on each plate, unhealthy one squirrel baby was left out and received no nuts at all. (So there was one sad little squirrel now.)

But when she tried putting 5 nuts per plate, cialis there was one nut left over. (At least nobody was unhappy this way.)

What is the sum of the number of the nuts and the number of squirrel children?


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

 

Fireball Frank

His real name was Frank Johnson, but everybody who followed baseball in the Tri-County Baseball League just called him “Fireball”. He was definitely the best pitcher on his team, the Bensonville Beagles, and probably the best in the whole 10-team league. He earned that nickname, and the respect of every batter who had to face him, due to his blazing fast ball pitch. Legend has it that the ball was nearly invisible as he threw it to the plate.

One day the Hatcherton Hawkeyes came to play on Bensonville’s home field. Fireball was going to pitch for the Beagles. The heckling got started early, led by Big Mark Stockdale, a pretty fair hitter of home runs on the Hawkeyes’ team.

“Hey, Fireball,” goaded Mark, “you think you’re a hot pitcher, I’d guess. You throw pretty hard to your catcher.”

“Well, reckon I do, ’cause he complains of a sore hand after games I pitch.” said Frank.

“So you’re good throwin’ to a batter. But just how good are you at throwin’ upwards to the sky?” was Mark’s challenge. “Straight up vertically, I mean. How long do ya’ think ya’ could keep the ball in the air?”

“Never really thought about that,” replied Frank. “Maybe 4 or 5 seconds, I s’pose.”

“I bet you a 20-dollar bill you can’t keep it up in the air for more than 5 seconds. What do ya say about that, Fireball?”

“I say take out your money, Mark.” Frank said.

Just then an umpire stepped up and said, “I’ll hold the money, boys. And I’ve got a stop watch here to do the timing.” Everyone agreed to that. The umpire tossed Frank a new baseball.

Frank walked confidently out to the middle of the infield, took off his cap, and tossed it about 6 feet away from where he was standing. “The ball will come down and land in my cap,” he boasted.

Frank gave the ball a good rub to soften the leather a bit. He went into his windup routine, then turned sideways and with a mighty effort, sent that ball skyward. Up, up it went. Quickly it was no larger than a dot in a “dot-com” address. Then just as quickly, it returned to earth – landing right in Frank’s cap! What a pitcher!

When the umpire announced the time, the Beagles fans went wild with happiness. Their Fireball had done another great performance! But as Frank reached out to receive the $40 from the umpire, he felt a strange pain in his pitching arm. “Gee, I hope I haven’t hurt my arm throwing that way.”

Your task is to tell me how long the ball was in the air if Frank released the ball from his hand 2 meters from ground level with an initial upward velocity of 30 meters per second. (Round your answer to the nearest 10th of a second.)

 


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

And some technical help can be found in the Commentary for the problem at this site: Motorcycle Daredevil.

 

Time Is Power!

While taking a coffee break, my secretary, Sue, happened to glance at her digital watch. It showed the following time:

1 : 4 4

“That’s rather curious,” she thought. “If I remove the two dots, I’ll have the number 144, which is the square of 12. I wonder how likely that sort of thing happens during my workday?”

Assuming that Sue works from 7 a.m. until 7 p.m., what is the probability that her watch will show a square number? Express your answer to the nearest hundredth of a percent.

EXTRA: what is the probability that a cube, 4th power, 5th power, etc. will occur in addition to squares?

SUPER-EXTRA: what is the probability for a power of any kind appearing on a watch set to a 24-hour day? (This means, times can range from 0:00 to 23:59. And “first” powers are not allowed throughout this problem.)

HYPER-EXTRA: if we now use the “seconds” digits that appear on many watches, what is the largest square number that can occur? Largest cube?

Omega Numbers

A popular problem in mathematics classes about problem solving concerns finding the unit’s digit of a large power of a number. An example of this might be:

Find the unit’s digit of 24000.

Of course, the student solving this is not expected to compute 2 used as a factor 4000 times. The reasons should be obvious. Rather the solver begins by looking for patterns, and armed with that knowledge, deduce the answer in a simple, straight-forward manner.

We propose now the following variation on this theme:

State the 2-digit number formed
by the final pair of digits of 22004.

Explain your process clearly, with enough data to establish your claim.


Please note: use of a simple calculator (with 8- or 10-digit displays) is permitted, however, such computing aid is not really even necessary. What is not permitted is the use of high-powered computing software, such as Mathematica.

Portioning Out Peanuts

I have three younger brothers, buy cialis whose names are (in order by age) Gary, ailment Corky, shop and Steve. Though we have our natural differences, we do agree on one thing: we love to eat salted peanuts while watching football on TV!One Sunday I brought out a big bag of delicious salted, roasted peanuts to share with them. Being the math “guru” of the family, I decided on the following unique way to portion out the peanuts. I gave one-third of the quantity to Gary, one-fourth to Corky, and one-fifth to Steve, keeping the remaining portion for myself.

Though I received what was left after giving to my brothers, that does not necessarily imply that my portion was the smallest.

How did my portion rank in size, from most to least (1st, 2nd, 3rd or 4th)?


Extra: If I received between 50 and 60 peanuts as my share, how many peanuts were in the bag?

Then I’d Rather Not Know

Mrs Holmann could not take it any longer that everyone kept asking her about her son’s age. This time she made her answer complicated enough for people never to ask again… neither her son’s age, nor her own! She said: “If you square my age and subtract the square of my son’s, the number you get is exactly 2 times the square of the difference between our ages. Also, if you square his age and you add it to the square of mine, the number you get is exactly 7776 less than 4 times the square of the sum of our ages!”

Can you work out how old they are?

[Note: this problem was written by Zsuzsanna Sukosd, a 15-year-old high school math student from Denmark.]

Calculating Expressions

One evening Elisa was doing her math homework when Dina came by to visit. “The exercise I have to do,” said Elisa, “is this one.”

Convert this expression to its calculator key-sequence form, then use your calculator to find its value.

“So far I have this much done,” she added.

12 [×] 5 [-] 8 [÷] 4 [+] 7 [×] 2 [=]

“That looks fine to me,” said Dina. “Let’s evaluate it now.”

Each girl took her own calculator and confidently entered the numbers and symbols as Elisa had given them. But their smiles quickly turned to a puzzled look on their faces when they realized that different results came up.

“How can that be?” they asked almost simultaneously. “Let’s do it again. Maybe we pressed a key incorrectly.”

Once again they entered the expression, only to have the first results to be confirmed. “Ah, I think I know the trouble,” said Dina. “Let me see your calculator.”

Elisa showed her this:

Dina then said, “Here’s mine.”

You see, it’s obvious now. They were using different kinds of instruments. So your task for this POW is to state what value each girl obtained and explain why it happened, based on the type of calculators being used.


Calculating Expressions

Trotter’s Treats

     Legend has it that long ago, one of my ancestors was a master candy maker. His specialty was a unique item made of chocolate, fancy nuts, cream filling, and other ingredients that were a highly guarded family secret. He sold his product under the simple name of Trotter’s Treats, or simply TT’s. Demand was always high for his confection, and rumor is that he made a lot of money – for those times – selling it.

     This ancestor also had an eccentric quality about himself and how he promoted his candy to the consumers. You see, he sold it only in little boxes of 4 treats or 7 treats per box. That’s right, just 4 or 7. It didn’t matter to him. Take it or leave it, he always said.

     The people didn’t mind either, as long as they were buying for themselves and their immediate families. The trouble arose when someone who was planning a party for example, and wanted to buy an exact number of the candies in order to give each guest exactly one of the TT’s. They had to calculate carefully about how many of each size to buy.

     For example, for a party of 30 persons, the host or hostess could buy 4 boxes of 4 treats and 2 boxes of 7 treats. The math looks like this:


4 boxes x 4 treats = 16 treats

2 boxes x 7 treats = 14 treats

     And 16 + 14 = 30. That’s how they did it. Simple, don’t you agree?

     Over the years the people showed great interest in calculating just how many boxes of each size would be needed to make any given number of TT’s. Sometimes they even found more than one way the purchase could be made, but other times, to their great puzzlement and wonder, they found certain numbers could not be exactly produced.

     For example, no one doubted that exactly 10 treats could not be purchased. Two boxes of 4, giving 8, were too few, while any other additional box of 4 or 7 would be too many. Likewise, one box of 7 was too few, and an additional box of 4 gave 11.

     So the challenge then became: what was the largest number of TT’s that could NOT be purchased? Can you find that number?


     Later on, this wily old chap decided to change things a little. As the economy of his time was expanding, he decided to increase the number of treats in the small and large boxes to 5 and 8. The townspeople saw this as merely a new challenge. The debate then became: what was the largest number of TT’s that could NOT be purchased now?


     Eventually, the math teachers in the town began to realize that there was a great opportunity here to involve their students in some good problem solving. So they began asking their students to try different box sizes, like 3 & 5, 4 & 9, 6 & 7, and so on. The students then prepared reports on their investigations. Very interesting results were found. Some students even found a formula!

     So how about you? What can you do?

     Good luck, and let me know.

No Primes Here

     Given three different digits, it is easy to form all 6 possible 3-place numbers with them. Depending on the specific digits chosen, some of those numbers might be prime numbers, whereas others are composite. Take this case: 2, 3, and 5.   523 is prime, but 235, 253, 325, 352, and 532 are composite.

     (It is even possible that all could be composite, but that’s another story.)

     A similar situation occurs even if you have a pair of like digits and one other digit. An example might be 2, 3 and 3.  233 is prime whereas 323 and 332 are composites.

     But consider the case of two pairs of similar digits. For example, a pair of 2’s and a pair of 3’s. What can be said about all possible 4-place numbers?


     Extra: Generalize this by using a, a, b, and b instead of digits.

UP-down: Match 2

John has $120 in his bank account and saves $8 each week. Mary has $230 in her account and withdraws $6 each week. After how many weeks will they have the same amount? What will that amount be?


Are you having a sensation of “deja vú” just about now? That is, do you feel you’ve seen this before? Well, the answer should be: “yes” and “no”.

Yes: because the action of saving and withdrawing money is present once again. For the “no” part: at least the money numbers for the persons are different. (Then there is a slight change in the title of this page, if you noticed it. The word “Almost” must have some purpose or it wouldn’t be there, right?)

So, perhaps, just perhaps, the goal of equal balances can’t be reached this time. If you have considered this as a possibility, you are well on your way to being an intelligent and observant problem solver. This problem is therefore actually “more realistic” than the previous problem: “UP-down Match Game“. If you think about it, it is not too likely that if two people behave in such a way as described in that problem, that their balances would someday “come out the same!

In this new situation, as you will soon discover as you work on it, the balances do not become equal. However, they do come close to each other at some future point, “passing like ships in the night”.

Our chart strategy is again the order of the day, but this time it might be useful to employ an extra column, to see some important information as it develops.

Now we can alter our style of answer conclusion by stating something similar to this:

John and Mary will not have equal balances, but they will be very close in Week No. 8, where the difference is the smallest, $2.

Teacher Commentary

It is recommended that for this problem the students not be warned in advance that the money amounts do not eventually coincide. This is intentional with the hope that they will “discover” it in the natural course of doing the solving process. This little unexpected outcome is a good experience in watching the flow of data carefully and not assuming that just because the two problems appear to be the same, that they are indeed the same. This prepares them to “expect the unexpected” in math problems and to always be alert for subtle or great differences that might arise.

One of the main objectives of this lesson is to recognize the concept of “inequality”, and more specifically, when the two balances are unequal by the
least amount. In Week 7, John’s amount was less than Mary’s. But in Week 8, the situation was reversed! In fact, the absolute value of the differences was even smaller. This was just the way it happened this time; problems can be easily constructed in which the minimum difference occurs on the earlier week (see Appendix: #2 below).

[It should perhaps be noted that negative numbers were given in the chart. That was a result of always subtracting Mary – John during its construction. Whether a given class should utilize this concept should depend on the level of the students involved. It is not essential to this lesson’s topic as a whole. Certainly it does provide a natural situation for giving negative numbers an important use!]

After reflecting on the “tricky wording” of the problem as given at the top, it might be suggested for students to come up with a new wording that does not deceive” the reader. An example might be this:

John has $120 and saves $8 each week. Mary has $230 and withdraws $6 each week. After how many weeks will the amounts of money they have be the closest? And what is the difference between them?

Now that equal-amount problems and different-amount problems have been covered, students can now be encouraged to write their own. Writing one’s own creations had to wait until this point; it might have been a little difficult for young learners to directly write equal-amount problems any sooner. Now the element of surprise and wonder can enter the process. Whatever numbers are used, there is a meaningful outcome. If the balances match, good; if not, well that’s okay too.

This then brings up the next natural question: can one know in advance when type of outcome will result? The answer is Yes, but finding the “trick” is a problem to solve in itself.

Appendix

Here are a couple of extra problems to get you started. Answer data in red.

1. John: $234 saves: $6           w = 9 d = $10 $288 & $298
Mary: $370 withdraws: $8      w = 10 d = -$4 $294 & $290

2. John: $180 saves: $7           w = 11 d = $6 $257 & $263
Mary: $351 withdraws: $8        w = 12 d = -$9 $264 & $255

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