Choose any 4 digit number except for 1111, 2222, 3333, 4444. I’ll choose 1502.

Rearrange the digits to give you the biggest and the smallest numbers you can. 5210 and 0125.

Find the difference of these two numbers. 5210 – 0125 = 4995.

Repeat for the new number you get (using zeros to supplement any missing digits if necessary; you must always have 4). 9954 – 4599 = 5335.

Keep repeating until you have a good reason to stop.

Example

5210 – 0125 = 4995

5533 – 3355 = 2178

8712 – 1278 = 7434

7443-3447 = 3996

9963 – 3699 = 6264

6642 – 2466 = 4176

7641 – 1467 = 6174.

I’m not going to repeat any more because 6174 => 7641 – 1467 which is the calculation I just did!

Discussion

• So every number we’ve tested came to 6174. Do you think all numbers will come to 6174? Why? In maths we can’t say we’re sure about something unless we have a good reason, so unless you want to go through every number and check it, we can’t say that we know that every number sequence we choose will go to 6174! (This is how mathematics differs from science, we don’t just make hypotheses and wait until the next time they break, we find ways to be certain!)
• Would we have to check every different number to be certain or are there shortcuts we can take? (should we check both 1234 and 1243 separately?)
• My rules stipulated that you couldn’t choose all the same number: 1111 or 2222. What would happen if you did use those numbers?
• What about two digit numbers or three digit numbers?
• Could we frame the problem as (1000a + 100b + 10c + d) – (1000d + 100c + 10b + a)?
• What about five or six digit numbers (or more…)? You may need a computer to help!

Thoughts

I just came across this interesting bit of mathematics via Twitter: http://plus.maths.org/content/os/issue38/features/nishiyama/index. It offers the opportunity for a beautiful open-ended task for secondary maths classes of all abilities. The initial task is easy and produces a startling result that feels like a trick; that hook can lead to discussion and further work about probability, permutations, algebra and programming.

http://vimeo.com/1908224?hd=1

As the comments underneath the video allude to, there is some utility here to attempt to consider the size of magnification that occurs during a portion of the video, and how to express magnifications of such a magnitude.

Cell Size And Scale

From a scientific perspective it is interesting to reflect upon the relative sizes of elements, from a Coffee Bean to a Carbon Atom (through various things such as a human Ovum, Sperm, various viruses, compounds and so on).

From a mathematical perspective it is also interesting for the way in which the relative scales are measured in the top-left. Exploring the different notations for small sizes would be a useful exercise in place-value for all levels of Key Stage 3 and 4.

Visit Flickr for the original at a range of sizes.