House Number? (1)

Ok this is how I came up with the solution. (I'm sure there are multiple ways of doing this though).

Short Answer:

Summation of even houses beneath 420 (lowest house within tested range): 43890

Summation of even house numbers on street: 57360

Total summation (even) upper to lower limit: 57360-43890=13470

Hence the summation of even houses from my house to both the upper limit (478) and lower limit (420) can be found via equation:

$\frac { 13470-(2n+2) }{ 2 } =n(n+1)-43890$

where 'n' is the place number (along) of the house below mine.

This cancels to:

${ n }^{ 2 }+2n-50624=0$ with solution n=224

Hence the house below mine has number 2*224=448.

Therefore my house is number 450.

Long Answer:

To start with, lets designate the house number we want to find as λ. We know that there is a number of houses (even) between this house and house number 418. We also know that there is a number of houses (with the same even summing value) between this house and the end of the street at 478. Both are non inclusive.

Hence: 418 < λ < 478. We also know that λ is even.

An alternative way of thinking about this is to think how 'far along' the street these various houses are. House 418 is 209 places along the even side of the street, house 478 is 239 along.

We know that Σ even house numbers up to 478 subtracting the Σ up to and including my house (λ) is equal to Σ even house numbers up to and including house number λ-2 as long as we also take away the Σ of even houses below 420.

The value of the even summation beneath 420 (not inclusive) is calculated by taking the place number along of house 418 and plugging it into the formula for calculating even integer series: (n(n+1)).

Therefore: 209*210= 43890.

We also know the summation of even numbered houses of the street up to 478. Again take the place number (now 239) and the value turns out to be 239*240= 57360.

The summation of the even house numbers between 420 and 478 then is 57360-43890=13470.

Hence the summation value of both the even houses between 418 and λ (my house) and between my house and the end of the street, (Not including λ/my house or 418), is this:

$\frac { 13470-(2n+2) }{ 2 }$

Here 'n' is the place number of the house just below mine on the even side of the street.

This value is equal to the summation of even numbered houses between either λ and 418 or λ and 478 (including 478). Therefore we get the following equation:

$\frac { 13470-(2n+2) }{ 2 } =n(n+1)-43890$

This then follows the progression:

$13470-(2n+2)=2n(n+1)-87780$

${ 2n }^{ 2 }+2n+2n+2-87780-13470=0$

${ 2n }^{ 2 }+4n-101248=0$

After solving: n=224

As the value of n is 224, we know that the house just below mine is house number 448.

Therefore I have house number 450 . Hope that helped.