Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

Card Trick 2

Can you explain how this card trick works?

Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Reverse to Order

Stage: 3 Challenge Level:

In general, if $a$ and $b$ are single digits and $a > b$, $ba$ becomes $ab$.

Between the two numbers $ab$ and $ba$ there is a difference of

\begin{align} 10 ( a - b ) + ( b - a ) &= 10 a - 10 b + b - a\\ &= 9 ( a - b )\; . \end{align}

So to turn $ab$ into $ba$ do the following calculation:

$$ba = ab - 9 ( a - b )\;.$$

Here is an alternative solution:

Add the digits, multiply by $11$ and then subtract the original number.

Can you justify it?

Correct solutions to this problem were received from: Archbishop Sancroft High School; Stephen and Adrian - South Greenhoe Middle School; Jack, Paul and Matthew - Smithdon High School.

We also received a great solution by Emily from Stanley Park Junior School. She thought about how to reverse the order of the digits of any three or four digit number:

If we reverse the order of the digits of the three digit number $htu$ we obtain $h+10t+100u$.
Reversing the digit order of the four digit number $Thtu$ yields $T + 10h + 100t + 1000u$
($u$ = units digit, $t$ = tens digit, $h$ = hundreds digit, $T$ = thousands digit).

For example, to reverse the digits in the number $1234$ you would do the following:

$1 + 10\times 2 + 100\times 3+1000\times 4$ .
The sum of all the numbers is $4321$, the reverse of $1234$.