### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

### Fibs

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

# Square Grid

##### Age 11 to 14 Short Challenge Level:

Answer: $\frac{101}{200}$

Notice that there is a total of $100\times100=10 000$ little squares in the grid.

Cutting the grid in half
The red line cuts the grid in half.

On each row, there is half of a shaded square on the 'wrong' side of the red line.

So it is $\frac12$ shaded plus $\frac{\frac12}{100}=\frac1{200}$ for the half square per line.

$\frac12 + \frac{1}{200} =\frac{101}{200}$

Add one more row so that the new shape is half shaded:

Grid on right: $100\times101=10 100$ squares, $10 100\div2=5050$ are shaded.

Grid on left: also $5050$ shaded, out of only $10 000$
$\therefore$ fraction shaded is $\frac{5050}{10000}=\frac{101}{200}$

'Counting' the number of shaded squares by making full rows
Move shaded squares to make full rows (and empty rows):

$1$ square is moved into the row with $99$ shaded,
$2$ into the row with $98$ shaded
...
$49$ are moved into the row with $51$ shaded
$50$ are left in the row with $50$ shaded

Total $50\times100+50=5050$ shaded squares (including the row which already had $100$ shaded)

Fraction shaded: $\frac{5050}{10000}=\frac{101}{200}$

You can find more short problems, arranged by curriculum topic, in our short problems collection.