Missing Jigsaw Pieces

The puzzle has $8\times 6 = 48$ pieces.The number of possible combinations of missing pieces are $N = \left(\begin{array}{c} 48 \\ 3 \end{array}\right) = \frac{48\cdot 47\cdot 46}6 = 17\:296.$

If the three pieces are adjacent, they form either a straight line or an L-shape. The straight line can be horziontal or vertical; the L-shape can have four different orientations.

• Horizontal line: fits in a box of $3\times 1$ . There are $6\cdot 6 = 36$ possible positions.

• Vertical line: fits in a box of $1\times 3$ . There are $8\cdot 4 = 32$ possible positions.

• L-shape: fits in a box of $2\times 2$ . There are $7\cdot 5 = 35$ possible positions for each orientation, and with four different orientations that gives $4\times 35 = 140$ .

In total, then, $n = 36 + 32 + 140 = 208$ choices of three pieces result in a connected shape. The probability is $p = \frac n N = \frac{208}{17\:296} = 0.012\:025\:9\dots$ Therefore we report the answer as $\boxed{12026}$ .

It's easy to generalize for a $h \times l$ -size jigsaw puzzle. Following @Arjen Vreugdenhil solution, the probability $\mathbb{P}(h,l)$ that the three pieces form a connected shape is

$\displaystyle \mathbb{P}(h,l)=[h(l-2)+l(h-2)+4(h-1)(l-1)] \binom{hl}{3}^{-1}$

So, keeping the puzzle ratio of this particular one $(4:3)$ , we can solve

$\displaystyle \mathbb{P}\bigg(\frac{4}{3}l,l\bigg)=\frac{1}{10^6}$

giving $l \approx 67$ and $h\approx 89$ . Hence, in order to be right, the nephew should have solved a $5963$ -pieces puzzle!