puzzles, coin weighing, coding

[Van Vu <t-vanvu@microsoft.com>]

Hi ca?  lang,
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About 12 -13 coins problem: You can certainly find the bad coin among 13
as follows:

partition 4-4-5. weight 4 with 4 , if not equal than the bad coin is
here and process as in the 12 coins case. 

if equal than the bad coin is in the last 5. Take 3 of them and weigh
with 3 good coins (all coins
from the first weighing are good). If non equal than you know that the
bad coin is among 3 
and also know if it is lighter or heavier. Than finish as usual.  If the
3-3 weighing is equal than the bad coin is in the last 2. Take any of
them and weigh with a good coin and you are done.

I guess one cannot find the bad coin among 14, using 3 weighing. Can you
prove that ?
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-
100 Bao gao puzzle

Solution: take i bao gao from store number i. (i from 1 to 100) The
total weights of  5050 bao gao taken 
tells you which store is the bad one. 
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--------New puzzle:

Suppose you have a finite sequence of  natural number (integers >0) and
want to send it to a friend 
using one digit 1 only (meaning what you  send and what your friend gets
is a sequence of 1's). How can you encode  it so that 
your friend is able to read your original sequence ? 

Best, 
Van H. Vu