The quarter-squares are the numbers of form $k^2$ and $k(k+1)$.

Start by expressing $N$ as the sum of four squares.

If you used some square four times, i.e. $N=x^2 + x^2 + x^2 + x^2$, then $N=(2x)^2$ is a quarter-square.

If you used some square three times but not four times, i.e. $N=x^2 + x^2 + x^2 + y^2$, then $N=(x-1)x + x^2 + x(x+1) + y^2$ is a sum of four distinct quarter-squares.

If you used some square twice combined with two other distinct squares, i.e. $N=x^2 + x^2 + y^2 + z^2$, then $N=(x-1)x + x(x+1) + y^2 + z^2$ is a sum of four distinct quarter-squares.

If you used two distinct squares twice each, i.e. $N=x^2 + x^2 + y^2 + y^2$, then $N=(x+y)^2 + (x-y)^2$ is a sum of two distinct quarter-squares.

Finally, if you used four different squares, then trivially $N$ is a the sum of four distinct quarter-squares.

The quarter-squares are the numbers of form $k^2$ and $k(k+1)$.

Start by expressing $N$ as the sum of four squares.

If you used some square four times, i.e. $N=x^2 + x^2 + x^2 + x^2$, then $N=(2x)^2$ is a quarter-square.

If you used some square three times but not four times, i.e. $N=x^2 + x^2 + x^2 + y^2$, then $N=(x-1)x + x^2 + x(x+1) + y^2$ is a sum of four distinct quarter-squares.

If you used some square twice combined with two other distinct squares, i.e. $N=x^2 + x^2 + y^2 + z^2$, then $N=(x-1)x + x(x+1) + y^2 + z^2$ is a sum of four distinct quarter-squares.

If you used two distinct squares twice each, i.e. $N=x^2 + x^2 + y^2 + y^2$, then $N=(x+y)^2 + (x-y)^2$ is a sum of two distinct quarter-squares.

Finally, if you used four distinct squares, then $N$ is clearly the sum of four distinct quarter-squares.

IIUC, quarter-squares are numbers of form k^2 or k(k+1).

Start by expressing N as the sum of 4 squares. If you use some square four times (x^2 + x^2 + x^2 + x^2), replace it with (2x)^2.

If you use some square three times (x^2 + x^2 + x^2 + y^2), use (x-1)*x + x^2 + x(x+1) + y^2.

If you use some square twice (x^2 + x^2 + y^2 + z^2), use (x-1)x + x(x+1) + y^2 + z^2.

If you use two squares twice each (x^2 + x^2 + y^2 + y^2), use (x+y)^2 + (x-y)^2.

If you use four different squares, that solution works.

The quarter-squares are the numbers of form $k^2$ and $k(k+1)$.

Start by expressing $N$ as the sum of four squares.

If you used some square four times, i.e. $N=x^2 + x^2 + x^2 + x^2$, then $N=(2x)^2$ is a quarter-square.

If you used some square three times but not four times, i.e. $N=x^2 + x^2 + x^2 + y^2$, then $N=(x-1)x + x^2 + x(x+1) + y^2$ is a sum of four distinct quarter-squares.

If you used some square twice combined with two other distinct squares, i.e. $N=x^2 + x^2 + y^2 + z^2$, then $N=(x-1)x + x(x+1) + y^2 + z^2$ is a sum of four distinct quarter-squares.

If you used two distinct squares twice each, i.e. $N=x^2 + x^2 + y^2 + y^2$, then $N=(x+y)^2 + (x-y)^2$ is a sum of two distinct quarter-squares.

Finally, if you used four different squares, then trivially $N$ is a the sum of four distinct quarter-squares.