Marie agrees to trade her PC in exchange for peter's gold chain. The
chain initially consists of 63 gold links arrange inline (open chain).
Peter wants a guarantee equal in days to the number of links in the
gold chain. To facilitate the enforcement of the guarantee, he insist
on paying by the installment plan, one gold link per day until the end
of the 63 day period. Marie agrees to the request, but insists that
the installment payment be effected not cutting more than three links
in the gold chain. Can this be done, and , if so, what links in the
chain should be cut? The answer must be responded in the amount of
links, and chains reaming after the cuts and which link.

The solution to Problem #2 here may be helpful in formulating an answer:

http://www.answers.google.com/answers/threadview?id=363060

Peter cuts the 5th, 9th and 17th links of the chain, giving him 3
individual links and pieces of the chain with 4, 8, 16 and 32 links
each.

With this, he can give Marie 1 link for 3 days, then take them back in
change for the 4 link piece, and so on.

The problem is very interesting... i hope this will work :)

Cut the gold links at link nos 5, 14 and 31 giving 3 individual cut
pieces(CP) and short chains(SC) of 4 links [SC(4)], 8 links[SC(8)] ,
16 links[SC(16)] and 32 links[SC(32)]
   0000 C 00000000 C 0000000000000000 C 00000000000000000000000000000000  


DAY     ACTIVITY         
Day 01: Give 1CP    #Give 1 Cut Pieces :)
Day 02: Give 2CPs         
Day 03: Give 3CPs         
Day 04: Give SC(4)  #Give ShortChain wth 4 links SC(4) & take back the 3 Cut Pieces
Day 05: Give SC(4)+ 1CP   
Day 06: Give SC(4)+ 2CPs  
Day 07: Give SC(4)+ 3CPs  
Day 08: Give SC(8)        
Day 09: Give SC(8)+ 1CP   
Day 10: Give SC(8)+ 2CPs  
Day 11: Give SC(8)+ 3CPs  
Day 12: Give SC(8)+ SC(4)
Day 13: Give SC(8)+ SC(4)+ 1CP
Day 14: Give SC(8)+ SC(4)+ 2CPs
Day 15: Give SC(8)+ SC(4)+ 3CPs
Day 16: Give SC(16)
Day 17: Give SC(16)+ 1CP
Day 18: Give SC(16)+ 2CPs
Day 19: Give SC(16)+ 3CPs
Day 20: Give SC(16)+ SC(4)
Day 21: Give SC(16)+ SC(4)+ 1CP
Day 22: Give SC(16)+ SC(4)+ 2CPs
Day 23: Give SC(16)+ SC(4)+ 3CPs
Day 24: Give SC(16)+ SC(8)
Day 25: Give SC(16)+ SC(8)+ 1CP
Day 26: Give SC(16)+ SC(8)+ 2CPs
Day 27: Give SC(16)+ SC(8)+ 3CPs
Day 28: Give SC(16)+ SC(8)+ SC(4)
Day 29: Give SC(16)+ SC(8)+ SC(4) + 1CP
Day 30: Give SC(16)+ SC(8)+ SC(4) + 2CPs
Day 31: Give SC(16)+ SC(8)+ SC(4) + 3CPs
Day 32: Give SC(32)  
Day 33: Give SC(32) + 1CP          
Day 34: Give SC(32) + 2CPs         
Day 35: Give SC(32) + 3CPs         
Day 36: Give SC(32) + SC(4)    
Day 37: Give SC(32) + SC(4)+ 1CP
Day 38: Give SC(32) + SC(4)+ 2CPs
Day 39: Give SC(32) + SC(4)+ 3CPs
Day 40: Give SC(32) + SC(8)
Day 41: Give SC(32) + SC(8)+ 1CP
Day 42: Give SC(32) + SC(8)+ 2CPs
Day 43: Give SC(32) + SC(8)+ 3CPs
Day 44: Give SC(32) + SC(8)+ SC(4)
Day 45: Give SC(32) + SC(8)+ SC(4)+ 1CP
Day 46: Give SC(32) + SC(8)+ SC(4)+ 2CPs
Day 47: Give SC(32) + SC(8)+ SC(4)+ 3CPs
Day 48: Give SC(32) + SC(16)
Day 49: Give SC(32) + SC(16)+ 1CP
Day 50: Give SC(32) + SC(16)+ 2CPs
Day 51: Give SC(32) + SC(16)+ 3CPs
Day 52: Give SC(32) + SC(16)+ SC(4)
Day 53: Give SC(32) + SC(16)+ SC(4)+ 1CP
Day 54: Give SC(32) + SC(16)+ SC(4)+ 2CPs
Day 55: Give SC(32) + SC(16)+ SC(4)+ 3CPs
Day 56: Give SC(32) + SC(16)+ SC(8)
Day 57: Give SC(32) + SC(16)+ SC(8)+ 1CP
Day 58: Give SC(32) + SC(16)+ SC(8)+ 2CPs
Day 59: Give SC(32) + SC(16)+ SC(8)+ 3CPs
Day 60: Give SC(32) + SC(16)+ SC(8)+ SC(4)
Day 61: Give SC(32) + SC(16)+ SC(8)+ SC(4) + 1CP
Day 62: Give SC(32) + SC(16)+ SC(8)+ SC(4) + 2CPs
Day 63: Give SC(32) + SC(16)+ SC(8)+ SC(4) + 3CPs

Regards.
yäiphä

YUp, I made a mistake counting.