IBM Ponder This #030 · October 2000

This puzzle is due to a suggestion by Denis Borris.
(Thanks, Denis.)

Part A:
There are three participants: a Host, a Partner, and a Volunteer.
The Partner is in a soundproof room.
The Host gives the Volunteer six blank cards, five white and one blue.
The Volunteer writes a different integer from 1 to 125 on each card, as the Host is watching.
The Volunteer keeps the blue card.
The Host arranges the five white cards in some order and passes them to the Partner.
The Partner then announces the number on the blue card.
How?

Part B:
Same set-up, except:
The integers now range from 1 to N, where N=190.
The Host can either pass all five white cards, as before, or discard one and just pass four cards.

Part B is much harder than Part A. Special mention will be given to the first person to give a justifiable procedure that works when N is at least 190; and to the first person to give a justifiable procedure for the largest value of N submitted (even if that's smaller than 190).

Notes:
There are no tricks like upside-down cards; the only information that the Partner gets is the cards and their ordering.

Solution

Best opened after a real attempt

To be added.