Friday, May 28, 2010

MathNotations Soaring With Eagles or Just For the Birds? Updates 5-28-10

NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.

I'll get to that cryptic title in a moment (may be obvious to some)...

1.  Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:

Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.

Mr. Lomas: 5! - (2+2)! - 1 - 0   Perhaps the most elegant since it uses the individual digits in the given order.

Robot Guy: (21-2)*5+0

Nate (high schooler): 120-5^2   Oh, the simplicity of that one! Combining digits is not the first way I thought of...

Mine so far:

(a) 102 - (5+2)  Pretty simple but I wasn't thinking much of combining digits until I saw Nate's

(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)

(c) (2^5)(2+1) - 0! (I posted this one already)

(d) 10^2 - 5 x (2 - 0!)   (I knew there had to be a way using 100 - 5)

(e) A new one: (2 + 2)! x (5-1) - 0!  I felt I needed to atone for my error in (d)!

I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!

Keep these coming if you can find more. I'd like to see us get to 10 ways.

2. Remember the hens -a- layin' problem I posted a few days ago? The video on YouTube gave the answer for 6 hens in 6 days: 24 eggs.

The problem on the blog was:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Here the answer is: 6 eggs

Here's a black-box method, i.e., work shown but no explanation:

(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.
This is how most solutions are given online and in the literature. It has little to do with middle schoolers actually learning the underlying principles. See the video for details.

3. Now for something completely different as M.P would say!
I've decided for now to tweet a daily (SAT) Problem of the Day.  "SAT" is in quotes because you can use these in your class as regular warm-ups or students can try these on their own to prepare for the upcoming SAT on June 5th and beyond.
Answers to each question will generally appear the next day, just before I tweet the new question. I've posted two problems thus far and the answers are up there today. Today's question will appear shortly.

My Twitter address is naturally dmarain.
Get the RSS feed for this at Twitter/dmarain if you want to see the daily problems.
If you have a question about the problems or want more details about solutions, send me a Direct Message in Twitter or email me.

Follow me if you'd like. These questions will not appear on this blog, so you will need a Twitter account or subscribe to the RSS feed above. Let your students know about it as well if you'd like.

Let me know by commenting here or replying on Twitter (Direct Message) if you like these and want me to continue next fall. Last SAT Problem of the Day on Twitter for this school year will be 6-15-10.

Requiescant in Pacem, Martin...

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific


three of clubs said...

(0+1)| (2x2)!-5! |

Dave Marain said...

three of clubs--
I like that form but check the arithmetic. If (0+1) is multiplied by the absolute value then I obtain 96. I think all you need do is to take your expression in abs values and then subtract (0+1) yielding 96 - 1 = 95.
Regardless, you found a very interesting one.

Anonymous said...


there's an extra zero in your example d.

Not very original, but with a twist:

[(2 - 0!)/.1]^2 - 5


Dave Marain said...

thanks for pointing out my error!
Naturally your decimal approach makes me think of using repeating decimals:

2 / (.02 repeating) + 1 - 5 = 2 / (2/99) -4 = 99-4 = 95.
I owe that one to you Jonathan! I hope it makes up for my carelessness.

These kinds of number puzzles do take on a life of their own and there are so many popular variations of these kinds of problems. I really do think they improve student's number sense and review order of operations.

In actual practice, do you think most students would have calculator in hand while playing with this riddle? How many of your students do you think would attempt to do this with paper and pencil and some mental calculation like we do! I'm sure there are some...

Dave Marain said...

I just added an extra one in the original post, labeled (e). Hopefully I've now atoned for "my sin"!! (wasn't My Sin once a perfume by some French company?).

Here it is again:
(5-1) (2+2)! - 0!

Is there any end to this? Actually, the issue of how many possible expressions one could make seems to be a formidable challenge, even if we place restrictions on the operations.

gremio said...

I do believe there are limits if you restrict the operations adequately.

With just + - * / ^, I don't think there are any others besides

120 - 5^2

(2-21) * (0-5)

102 - (5+2)
102 - 5 - 2


In terms of limits in general, with binary operations you always consume an input, so the search depth is limited by that. If you add in unary operations like sqrt or factorial, then you can deepen quite a lot more, though perhaps not usefully. Other unary operations like floor and ceiling only apply once non-redundantly, so those don't matter.

Other binary operations I'd try would include modulo, log with a specified base, nth-root... what else?


Dave Marain said...

Nice analysis! I knew one of my readers would be able to see a few levels into this. Along the way, your systematic approach produced a few other nice combinations. Thanks! Now how would you write a program in Python to analyze all the possibilities!

By the way, I'm sure I could have picked almost any other date and someone's age and we could have generated many solutions. But I just get the feeling there's something special about Martin Gardner's legacy to us...