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Summing Consecutive Numbers

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1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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.

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

Adding another row
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.