Magical Products
Can you place the nine cards onto a 3x3 grid such that every row, column and diagonal has a product of 1?
Put the nine numbers below onto a 3 by 3 grid so that each number occupies one space, and the product of every row, column and diagonal is equal to 1.
$$1 2 3 6 \frac{1}{6} \frac{1}{3} \frac{1}{2} \frac{2}{3} \frac{3}{2}$$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This is one answer (seven others are possible by rotation and reflection)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525207.png?itok=gmp8b-Sy)
Put the 1 in the middle so that reciprocals can go opposite each other.
Try placing the 6 at a corner:
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525201.png?itok=NwuzVYth)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520producst%2525202.png?itok=j6GhVz2D)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525203%252520corrected.png?itok=tJ6d_Rqc)
But $\frac13$ is opposite $\frac16$ and there is nothing large enough to multiply by to get $1$
$\frac12\times\frac16$ is also too small so swapping $\frac12$ and $\frac13$ would not help
Try placing the $6$ in the middle of an edge:
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525204.png?itok=ksl6xT8u)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525205.png?itok=_C8kgX47)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525206.png?itok=yRRkz1i5)
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![Magical Products Magical Products](/sites/default/files/styles/large/public/thumbnails/content-id-7189-magical%252520products%2525207.png?itok=gmp8b-Sy)