Checking how many inputs (reed switches) are active using minimal GPIO pins
As mentioned by others, it can be done using an 8x8 matrix. 
If you look at the internet you will find plenty of picture where they are used for keypads.

However those standard schematics are assuming you press only one key at a time. Some combination of two switches are also possible. But in your case need to be able to detect any combination if pressed switches. 

I worked on that problem a while ago and found you can do it with an 8x8 matrix, but, as also mentioned by  WhatRoughBeast, it requires a diode in series with each switch.