000RM.dvi

514 Combinatorial games

Exercise

1.What are the winning positions in the battle of numbers withN=

100 andd=7?

2.The golden ratioφhas the property thatφ^2 andφhave exactly the

same digits after the decimal point. Find all numbersθwhich have

exactly the same digits asθ^2 after the decimal point.

3.A announces a two digit number from 01 to 99. B reverses the

digits of this number and adds to it the sum of its digits and then

announces his result. A continues in the same pattern. All num-

bers are reduced modulo 100, so that only two digit numbers are

announced. What choices has A for the initial number in order to

insure that B will at some time announce 00?^2

2 56, 80 or 68.