# Sunday Times Teaser 2815 – PIN in the Middle

*by Nick MacKinnon*

#### Published: 4 September 2016 (link)

For Max Stout’s various accounts and cards he has nine different 4-digit PINs to remember. For security reasons none of them is the multiple of another, and none is an anagram of another. He has written these PINs in a 3-by-3 array with one PIN in each place in the array. It turns out that the product of the three PINs in any row, column or main diagonal is the same. In fact there are just two different prime numbers that divide into this product.

What is the PIN in the middle position of the array?

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Here is a solution based on an analytical construction of multiplicative 3 by 3 magic squares comprised of integers.

Here is an alternative solution that does not assume a specific form for the magic square and is much slower (~ 10 seconds)

@Brian,

“Assuming that the 4 digit PINS are 4 digit integers, then a, b, c and d will all be in the range 10 to 99 inclusive.”

The range could be more optimal: 32-99 due to the squares in the edges.