This sly little trick will get you every time .
It's amazing how many ways folks come up with their own different solutions . . .

~ all wrong, of course ~

Make the six nails balance on the head of the seventh (in base) without touching anything below the head

\/ One THEORY could be . . . \/
(My son Richard was the first to adopt the following theory)

However, to execute this perfectly, all the nails have to be 'perfectly' straight;   the embedded one has to be 'perfectly' perpendicular to the base (assuming the base is 'perfectly' level to the surface upon which it rests), and its top 'perfectly' flat (no embossed imperfections) .
(it's a rare occasion when an imperfection (a burr,an embossed blemish,a defect in relief, etc.) can actually be an aid to the stacking, wherein the defect in relief can restrain the nail from rolling off the top, if it had that tendency to do so)

If any ~ or all ~ the nails are even slightly bent, the odds of success are extremely slim, although I'm sure it can be done if you have a great deal of time on your hands . . . (and that much more patience!) .

I'm sure perserverance would eventually win out .

But, there is a much simpler ~ and exponentially quicker ~ way to do it .
This method cares not about imperfections or slight bends in the nails . . .
. . . the top of the embedded nail in the base can actually be at a slight tilt!

~ all done within a minute ~

With forefinger and thumb, carefully grab the bottom horizontal nail ( along with the rest of the assembly) . . . . .
. . . . . and lift off the flat surface . . . . .
~ the ends of the inner four will quickly slack downwards, but will stop at a certain point ~
. . . . . and gently place onto the head of the 7th nail .

Important:   the weight of the top horizontal nail, and the equal position of all the side nails, will keep the assembly 'in balance' throughout the procedure from the time you pick up the whole assembly, to the time it's placed on top of the 7th nail's head .   Once some slack begins, or the assembly is tilted, the inner nails will lose their positions within the assembly, and everything will just fall apart .