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# Getting Round the City

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Age 11 to 16

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Annette from Copleston Sixth Form explains why there are $56$ possible routes from $A$ to $B$:

All routes between $A$ and $B$ will be equal in length as you need to

travel $3$ east and $5$ north in any order. In total there are $56$ different

paths leading to $B$.

The number in each square represents the number of possible paths from $A$ to that square.

It is worked out like this:

because you must arrive at a square from the point either directly south or directly west of it.

Leah from St. Stephen's School Carramar found a general rule for the number of ways of travelling from point $(x_1,y_1)$ to $(x_2,y_2)$ going only north and east and discussed some ways to find the number of paths from $A$ to $B$:

If the route must be taken without crossing diagonally, it will take $(x_2-x_1) + (y_2-y_1)$

units to reach the opposite point.

E.g the points $(0,1)$ and $(3,6)$, a path will be $(6-0)+(3-1) = 8$ units long.

Number of paths from $(x_1,y_1)$ to $(x_2,y_2)$ = number of paths consisting of $(x_2-x_1)$ steps east and $(y_2-y_1)$ steps north, in some order

$ =$ ${(x_2-x_1)+(y_2-y_1)}\choose{(x_2-x_1)}$ $= \frac{[(x_2-x_1)+(y_2-y_1)]!}{(x_2-x_1)!(y_2-y_1)!}$

How many paths are there from $A$ to $B$?

$\frac{(5+3)!}{5!3!} = 56$.

Alternative methods :

- Drawing a tree diagram via labeling each node as a

letter, which gives $56$ options.

-Drawing out all the possible routes.

Hannah from Burntwood explains how many different paths we can take if we only want to travel $6$ blocks:

A good way to think about this is as a tree diagram.

At the starting point, you have $4$ options of where to go.

Then at each subsequent point you have a choice of $3$ paths (assuming you can't double back on yourself)

So, the formula would be $4 \times 3^{(n-1)}$, where $n$=the number of blocks you walk.

In this case, $n=6$, so there are $4x3^5= 972$ paths.

Thank you to everyone who submitted a solution to this problem!

All routes between $A$ and $B$ will be equal in length as you need to

travel $3$ east and $5$ north in any order. In total there are $56$ different

paths leading to $B$.

$1$ | $6$ | $21$ | $56$ ($B$) |

$1$ | $5$ | $15$ | $35$ |

$1$ | $4$ | $10$ | $20$ |

$1$ | $3$ | $6$ | $10$ |

$1$ | $2$ | $3$ | $4$ |

$A$ | $1$ | $1$ | $1$ |

The number in each square represents the number of possible paths from $A$ to that square.

It is worked out like this:

$M$ | $M+N$ |

- | $N$ |

because you must arrive at a square from the point either directly south or directly west of it.

Leah from St. Stephen's School Carramar found a general rule for the number of ways of travelling from point $(x_1,y_1)$ to $(x_2,y_2)$ going only north and east and discussed some ways to find the number of paths from $A$ to $B$:

If the route must be taken without crossing diagonally, it will take $(x_2-x_1) + (y_2-y_1)$

units to reach the opposite point.

E.g the points $(0,1)$ and $(3,6)$, a path will be $(6-0)+(3-1) = 8$ units long.

Number of paths from $(x_1,y_1)$ to $(x_2,y_2)$ = number of paths consisting of $(x_2-x_1)$ steps east and $(y_2-y_1)$ steps north, in some order

$ =$ ${(x_2-x_1)+(y_2-y_1)}\choose{(x_2-x_1)}$ $= \frac{[(x_2-x_1)+(y_2-y_1)]!}{(x_2-x_1)!(y_2-y_1)!}$

How many paths are there from $A$ to $B$?

$\frac{(5+3)!}{5!3!} = 56$.

Alternative methods :

- Drawing a tree diagram via labeling each node as a

letter, which gives $56$ options.

-Drawing out all the possible routes.

Hannah from Burntwood explains how many different paths we can take if we only want to travel $6$ blocks:

A good way to think about this is as a tree diagram.

At the starting point, you have $4$ options of where to go.

Then at each subsequent point you have a choice of $3$ paths (assuming you can't double back on yourself)

So, the formula would be $4 \times 3^{(n-1)}$, where $n$=the number of blocks you walk.

In this case, $n=6$, so there are $4x3^5= 972$ paths.

Thank you to everyone who submitted a solution to this problem!

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.