Note: There has been a revision in one of the properties of 17 below - I'm sure you already caught the error! Joshua caught another error involving the 4th powers - both have now been corrected.
For middle or high school students:
Number sleuths -- get into your detective groups. No calculators for the first 5 minutes.
You will have a total of 15 minutes to uncover as many interesting or fascinating facts as you can about the number 97.
One member of the group must record these and report back. Make two columns.
The first labeled: Discoveries found without the calculator.
The other column: Discoveries found with the calculator.
Ah but you're wondering what makes some fact interesting or fascinating. That's pretty subjective, right? Let's model one:
- Prime number
- Both of its digits are odd
- The product of its digits is prime
- Can be written as a sum of 2 consecutive integers: 17 = 9+8
- If its digits are reversed, the resulting number, 71, is also prime and the difference of 71 and 17 is 64 or 82; oh, and 8 just happens to be 1+7!!
Note: This is clearly incorrect - 3-point penalty!!
- Can be written as a sum of squares: 17 = 42 + 12
- Can be written as a difference of squares: 17 = 92 - 82
- Can be written as a sum of a perfect square and a perfect cube: 17 = 32 + 23
- Can be written as a sum of 2 consecutive fourth powers: 17 = 14 + 24; in fact, 17 is the least integer which can be written as the sum of two distinct nonzero 4th powers
- You can get your driver's license in some states on your 17th birthday (this is clearly the only fact that's interesting to an adolescent - I had asked one of my students what's special about March 14 and he replied, "It's my birthday and it's also the Junior Prom.")
- 17 is the hypotenuse of a Pythagorean triple; 8-15-17 [high school level?]
- 17 is part of another Pythagorean triple: 17-144-145 [high school level?]
- Is there any end to this list?
You will have 15 minutes to complete your investigation of the number 97.
To rate each of your lists, we can assign a point value to each fact. For example, the first four facts listed above for the number 17 could each receive one point. Some of the other facts could receive 2, 3, 4 or even 5 points (max) for being more difficult to find or just more amazing. If any fact is incorrect, 3 points are deducted, so you'd better do independent fact-checking on your team! We will then determine the top 3 lists from this rating system and those teams will receive worldwide recognition by having their results appear on You Tube (just their lists, not names or faces for reasons of confidentiality of course). This will surely go viral in 17 seconds or less. Oh, so you'd rather receive 97 bonus points on your next test?
Oh, I forgot to mention. As a super extra credit bonus project/assignment, do the same this evening for the number 153. You will need to email me with your individual lists by 10 PM (yes, yes, we all know this is unfair to students who do not have access to email or whose parents may not give permission for this or for some other reason, so this is just an option!).