Transformation
In the grids above, two squares are defined to be neighbors if they share a side (not just a vertex). Some examples are
- 2 and 3 in Figure 1,
- 7 and 5 in Figure A, and
- 8 and 2 in Figure B.
Now, define operation as follows: choose any two neighboring squares and increase/decrease both values by the same integer.
Then which of the following statement(s) is/are correct?
I.
Figure 1 can be transformed into Figure A by iteration of
II.
Figure 1 can be transformed into Figure B by iteration of
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1 solution
Color the 9 squares in the above manner. Let R and be the sum of the numbers in the colored squares. Let S and be the sum of the numbers in the uncolored squares.
By iterations of Q , the values of R and S may be different but the value of the different, R − S , is fixed, and is equal to 5.
In
Figure B
, it is obvious that
R
−
S
=
5
and hence
Figure 1
CANNOT
be transformed into
Figure B
by iteration of
Q
.
In Figure A , R − S = 5 . Figure 1 CAN be transformed into Figure B by iteration of Q . Give it a try!