Jeremy has a box of crayons sitting on a shelf in a very dark room - too dark to distinguish crayon color. Inside this box, there are 21 red crayons, 13 blue crayons, and 6 brown crayons. Now, Jeremy would like to go into the room and grab some crayons out of the box, without taking the whole box. But remember, the room is very dark. And the light doesn't work. And Jeremy's flashlight batteries are dead. And he has no replacement batteries. And he's not allowed to use matches....In other words, Jeremy will have to grab the crayons without seeing what colors they are. Well, Jeremy is a very inquisitive boy, and this quest gets his restless mind working. "How many," he asks himself, "would I have to grab, if I want to make sure I get at least 2 crayons of the same color?" He thinks about this for a minute. "Obviously, I could just grab every crayon in the box, and then I'd know I have some of the same color." He shakes his head at the idea. "But I don't want to grab that many, because then they'll roll off the table, and I'll lose some... and of course little Jenny will eat some..." Jeremy pauses at the door to the dark room. "Then again, I could just grab 2 crayons and hope that they're the same color. But what if they aren't? I really want 2 crayons of the same color. Oh, if only someone could tell me the smallest number that I absolutely have to grab in order to guarantee I have at least 2 crayons of the same color! That's the important thing - I need a GUARANTEE. Surely it doesn't have to be the whole box, does it?" Question 1: What's the smallest number of crayons Jeremy needs to grab in order to guarantee that he has at least 2 crayons of the same color? And if you get that, here are a few more! Question 2: What if Jeremy wants to guarantee that he has at least 2 red crayons?Question 3: What if Jeremy wants to guarantee that he has at least 1 crayon of each color?Question 4: What if Jeremy wants to guarantee that he has more reds than browns?Solution available here. And if you still want more puzzles of a similar type, check out our "How Many Socks?" book, for sale at our store www.themathprofs.com. We'll walk you through the process of solving these types of questions - from questions just like these, slowly building to more general scenarios using variables for the numbers of colors and items.
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Step 1: Find the value of the clock in Row 4Note that the clock faces in Row 1 are NOT identical. That's the first mistake people will make, if they're speeding through the puzzle. They'll think the clocks are identical, and thus the sum of the three identical clocks is 21, which gives each clock a value of 7. But since the clocks are not identical, we have to find their value another way. Note that the clock faces are 9, 9, and 3, respectively. Now, what is the sum of \( 9+9+3\) ? Ah-ha!! There's our answer. Each clock has a value equal to the time showing on its face. So the clock in Row 4 is equal to 9. Step 2: Find the value of the calculator in Row 4If we look at Row 2, we see that this time the objects are identical. Thus, we can say 3 calculators is equal to 30, and therefore 1 calculator is equal to 10. But here we need to be careful, or else we'll make another mistake. To be more precise, we should have said that each of the calculators in Row 2 is equal to 10. You see, the calculator in Row 4 is not the same as the calculators in Row 2. The screen has a different number.So, what do we do? Well, it stands to reason that the value of each calculator is somehow related to its screen number. Does this work for the Row 2 calculators? We found that each Row 2 calculator has a value of 10, and each screen shows 1234. Well, what is the sum of \(1+2+3+4 \) ? Hey! I think we're onto something! If each calculator's value is equal to the sum of the digits showing on its screen, then that calculator in row 4 has a value of \(1+2+2+4=9 \). Step 3: Find the value of the three light bulbs in Row 4. Looking at Row 3, we see that all of the light bulbs are identical. So, on the left-hand side, we're simply starting with 1 light bulb, adding another bulb to it, and then removing the one we just added … which still leaves us with just one light bulb. This means that one light bulb has a value of 15, and therefore the 3 light bulbs in Row 4 are equal to....WAIT!! Did you catch it? I'll admit, I missed it the first time I looked at this problem. The light bulbs in Row 4 are NOT the same as the light bulbs in Row 3. The bulbs in Row 3 each have 5 light rays above each bulb. However, the bulbs in Row 4 each have 4 light rays. What does this mean? Well, it implies that the value of a light bulb somehow depends on the number of light rays above that bulb. We know that each bulb with 5 light rays has a value of 15. To me, this suggests that each light ray contributes a value of 3 to the total bulb value. So, if I want to know the value of a light bulb, I just count up the number of light rays above the bulb and multiply by 3. Since each bulb in Row 4 has 4 light rays, this means that each Row 4 light bulb has a value of \(4 \times 3 = 12\). And therefore, the 3 bulbs in Row 4 are equal to \(3 \times 12 = 36\). (Whew!! Now do you understand why I said this puzzle will test your observational skills, rather than your mathematical abilities?) Step 4: Putting everything together.Okay, substituting for the clock, the calculator, and the 3 light bulbs, Row 4 becomes \(9+10\times36\) Note that the second operation is multiplication, and not addition. (This detail is easily missed by the eyes when glancing from symbol to symbol.) And the Rules for Order of Operations tells us the multiplication is performed first, before the addition. Therefore, Row 4 is given by \(9+360=369\). Afterword:Okay, so 369 is our preferred solution. Now, usually in mathematics we don't speak of a preferred solution. We simply say the solution. But, unfortunately, for a lot of the viral puzzles floating around on social media, the rules are a little....vague. Maybe this helps make puzzles a bit more interesting (we're being generous, here), but it definitely creates some problems. Instead of being able to say "This follows from this, which follows from this, and therefore the answer is this," we are reduced to saying, "I think the best answer is given by this, because I think this is what the puzzle creator intended."Now, I want you to go back to our solution, and look at the assumptions we made. I can see several. For instance, when we noticed that the clock faces in Row 1 were different and also added up to 21, we quickly assumed the puzzle creator intended for clock values to be based on the number shown by the clock face. I think that's a pretty reasonable assumption, don't you? And I think most people would agree. But at the same time, the puzzle creator might have just put different numbers on the faces for variation, and really meant for all clock values to be the same no matter the face value. I doubt it. But it's possible. And that would lead to a different final answer, wouldn't it?But what about the calculator assumption? Once I saw that a calculator screen with 1234 was associated with the value 10, the first thought that occurred to me was "Oh, I just add the screen digits together and that gives the value." But others might have come up with a different scheme. And there really are infinitely many possibilities. I mean, basically you're just looking for a function that takes four input values (the four digits on the screen) and gives one output value. So the function will take the form \(f(x_1, x_2, x_3, x_4)=y \). Now, the only restriction on our creativity in building such a function is that a calculator screen with "1234" on it must have a value of 10. So our function must satisfy \(f(1,2,3,4)=10 \). The function I chose (where I simply summed up the screen digits) could be written as \(f(x_1, x_2, x_3, x_4)=x_1+x_2+x_3+x_4 \). But I can just as easily come up with a different function that still meets our criteria. What about \(g(x_1, x_2, x_3, x_4)=2x_2+2x_3 \)? Does that satisfy our condition? Indeed it does! Check it and you'll see that \(g(1,2,3,4)=2(2)+2(3)=10 \). And that function would yield a different final answer for our puzzle solution, right? In fact, do you realize that if you give me any number in the world, then I can make the puzzle solution equal to that number, simply by creatively choosing the function that evaluates the calculator screens?! And, since the rules are vague, all I would need to do is be willing to argue its defence in an authoritative manner, and I could claim that my solution is the correct solution, and everyone else is wrong - including the puzzle creator, if need be. But rather than go that route, I'd just stick with our preferred solution above, and most folks will agree with you without the need for added argument. Homework:And we can't leave you without some homework!! - Come up with a function \(f(x_1,x_2,x_3, x_4) \) so that \(f(1,2,3,4)=10\), and \(f(1,2,2,4)=23 \).
- Create a calculator screen function \(f(x_1,x_2,x_3, x_4) \) so that the solution to the entire puzzle is 100.
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A family member sent this puzzle to us. I think it may have been making some rounds on social media a while back. Be careful with it. Nothing too difficult mathematically, but it will certainly test your observational skills...so pay extra close attention!! And to be honest, one COULD make the case that there are infinitely many viable solutions to this one (that sounds strange, I know, and we'll explain in our solution). Regardless, we'll give you the solution we think is best and most defensible, and we'll show you how we would solve it on Friday. You can find the solution to this puzzle here. If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!
Your dear Aunt Sally wanted to check and see how you did on her homework assignment. Here are Aunt Sally's answers: - \(5 \div 5(1+4)=5 \)
- \(5(1+4) \div 5 \times 5=25 \)
- \(5 \times 5\div(1+4)=5 \)
- \( (5 \times 5) \div[5(1+4)] =1\)
- \(5(5 \div 5)(1+4)=25 \)
- \(5 \times 5\div[5(1+4)]=1 \)
If you struggled with any of them please let us know! If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!
When you evaluate the expression \( 5 \times 5 \div 5(1+4) \), which of the following is the correct answer? A) \(1\) B) \( 25\) C) \(125\) D) That's a poorly written math expression. E) A and D F) B and D G) C and D Believe it or not, the answer you give is determined by how well you know your Dear Aunt Sally!! Remember her? She's that stickler for rules that you probably met in third or fourth grade. Does PEMDAS ring a bell? When I was in school, we were taught to remember PEMDAS by thinking of it as an acronym for "Please Excuse My Dear Aunt Sally". The whole point of learning PEMDAS was to permanently stamp upon our brains the order of operations for arithmetic. Without rules for a precise ordering, any arithmetic expression becomes subject to dispute. There is no longer a certainty of meaning. Mathematical communication breaks down. Anarchy ensues. And then fights will break out, chalk and erasers will be thrown, etc. So, our Dear Aunt Sally is supposed to help us remember PEMDAS. And PEMDAS is supposed to help us remember...well...do you remember? Let's see, the "P" stands for "Parentheses". You always start with Parentheses. Then Exponents. Then Multiplication and Division, in order from left to right. And finally Addition and Subtraction, also in order from left to right. Now, take note. Because most people forget about "in order from left to right", and that causes problems. Lots of problems!! Some folks think that since "M" is before "D" in PEMDAS, then Multiplication is always performed before Division, wherever it occurs within the expression. But, NO! That's not correct. Multiplication and Division share the same hierarchy, and they are performed on a "first come, first served" basis, as you travel from left to right across the expression. Same goes for Addition and Subtraction. Let's look at the expression in our problem again: \(5 \times 5 \div 5(1+4) \). First we are going to work what is inside the parentheses. This gives us \( 5 \times 5 \div 5(5) \). Now, this is where many folks will mess up. They will automatically multiply those last two 5's together first, before doing anything else. Why will they do this? Well, often folks will do it because the multiplication is written using parentheses, and they're still thinking "Parentheses go first!". But the "P" in PEMDAS only says that what's inside parentheses goes first, not whatever is touching parentheses. Remember that \(5(5) \) really means \( 5 \times 5 \), so we could rewrite the expression as \( 5 \times 5 \div 5 \times 5 \). And now, here's where others will mess up. They're stuck thinking Multiplication always comes before Division, because "M" comes before "D" in PEMDAS. But recall what we said before. Multiplication and Division have the same hierarchy, so do them in order form left to right. So....let's do them in order from left to right. This gives\[ \begin{array}{c} 25 \div 5 \times 5 &=& 5 \times 5\\ &=& 25 \end{array} \] Okay, we see the expression evaluates to 25. But the answer is not (B). No, it's (F). Why? Because the expression really is poorly written! It's written as if...well, almost as if it was designed to trip people up. (Probably because it was!) In general, it's a bad idea to use the symbol \( \div \) in a math problem involving more than two numbers, unless you're prepared to make generous use of parentheses to remove any doubt as to intention. Writing mathematical expressions is just like writing sentences - the purpose is to convey ideas accurately and efficiently. Nine out of 10 people reading the expression above are going to breeze right through it and evaluate it incorrectly - unless they suspect it's a test, or some type of puzzle, and then slow down. So, yeah, most complex math problems don't involve the \( \div \) symbol. Instead they represent division using fractions. It's just clearer that way, at first sight. Hey, they're being nice to the speed readers! And we wouldn't be good math professors if we didn't assign some homework, of course. Here are some practice problems to help you remember your dear Aunt Sally. We will post answers tomorrow. - \(5 \div 5(1+4) \)
- \(5(1+4) \div 5 \times 5 \)
- \(5 \times 5 \div (1+4) \)
- \((5 \times 5) \div [5(1+4)] \)
- \(5(5 \div 5) (1+4)\)
- \(5 \times 5 \div[5(1+4)] \)
Solutions to the homework problems can be found here. Welcome to the Math Profs' Chalkboard! We hope to use this "chalkboard" to help you figure out math problems, solve puzzles, and find ways to add mathematics to your household. If you have any specific math questions or puzzles that you would like to see solved and analyzed, please use the contact form on our website or e-mail us at themathprofs@gmail.com. We look forward to hearing from you! ## Categories |
## AuthorsBrian and Melanie Fulton both earned doctoral degrees in mathematics at Virginia Tech. They formerly taught math at the university level, and now run a hobby farm while accuracy-checking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors. ## Archives## Categories |