In a very old book on my shelf it says:

Pick any three digits from 1 through 9 and write them down as a three-digit number. Let's call them a, b and c. For example if I chose 538, then a=5, b=3 and c=8. Now run through the following chain on your calculator:

a X 2 + 3 X 5 + b X 10 + c - 150

The answer will be the three digit number you wrote down.

a) Try this out. I did, and it did not work for me, did it work for you?

b) If not, assuming there is a typographical error or two, can you correct this?

c) If not, assuming there is no typographical error, what is going on?

d) Share the mental-jaunt you went on with someone - explaining what you were thinking at the very start and how you reached your destination.

To students (of all ages):

jaunt (noun) a short excursion or journey for pleasure

mental-jaunt (noun) a short mental excursion or mental journey for pleasure, nothing too heavy!

Halving, in mathematics, means to divide by 2. Halving is not "harfing".

Let's define "Harfing" as taking an integer with an even number of digits and separating them (centrally) into two digits (which are called the 'harfs')

e.g. the result of harfing 73 is 7 and 3 and the result of harfing 4580 is 45 and 80.

Can you find an integer for which squaring the sum of its harfs gives the integer?

If you find one can you find more?

If you do, what questions are you now wondering about?

Most importantly, will you share the mental-jaunt you went on - explaining what you were thinking at the very start and how you reached your destination.

Below you can download elements of one person's thinking, punctuated with reasoning why this activity is a positive one for students to explore.

m-j#1.pdf

m-j#1.xlsx