knights.gms : Maximum Knights Problem

**Description**

This MIP model finds the maximum number of knights that can be placed on a board. Two different formulations are presented. The second formulation is 'tight' and may perform better with certain MIP codes. Once we found the max number of knights, we solve a series of MIPs to find ALL solutions. We will use lags (relative positions) to describe the allowed moves. The labels H and V indicate horizontal and vertical moves as shown below: 0 0 0 0 X 0 0 0 0

**Reference**

- Dudeney, H E, Amusements in Mathematics. Dover, New York, 1970.

**Large Model of Type :** MIP

**Category :** GAMS Model library

**Main file :** knights.gms

```
$title Maximum Knights Problem (KNIGHTS,SEQ=158)
$Ontext
This MIP model finds the maximum number of knights that can be
placed on a board. Two different formulations are presented.
The second formulation is 'tight' and may perform better with certain
MIP codes. Once we found the max number of knights, we solve a series
of MIPs to find ALL solutions.
We will use lags (relative positions) to describe the allowed moves.
The labels H and V indicate horizontal and vertical moves as shown
below:
0 0
0 0
X
0 0
0 0
Dudeney, H E, Amusements in Mathematics. Dover, New York, 1970.
$Offtext
set i size of board / 1*8 /
n number of possible moves / m1*m8 /;
alias(i,j,k);
table move(*,n) all possible knight moves
m1 m2 m3 m4 m5 m6 m7 m8
H -2 -2 -1 -1 +1 +1 +2 +2
V -1 +1 -2 +2 -2 +2 -1 +1
variable total; binary variable x(i,j);
equations deftotal total knights on board
defmove(i,j) move restrictions
defmovex(n,i,j) move restrictions;
deftotal.. total =e= sum((i,j), x(i,j));
defmove(i,j).. sum(n, x(i+move('h',n),j+move('v',n))) =l= card(i)*(1-x(i,j));
defmovex(n,i,j).. x(i+move('h',n),j+move('v',n)) =l= 1-x(i,j);
model knight / deftotal,defmove /
model knightx / deftotal,defmovex /
option optcr=0,optca=.999;
solve knight use mip max total;
solve knightx use mip max total;
* Now we try to see how many different ways are there to arrange
* the same number of knights.
scalar maxknight; maxknight = total.l;
total.lo = total.l-.5;
option optcr=1,optca=100,limcol=0,limrow=0,solprint=off;
sets ss max number of solutions groups / seq1*seq20 /
s(ss) dynamic set for solution groups
parameter cutval all possible solutions for cut generation;
equation cut(ss) known solutions to be eliminated;
cut(s).. sum((i,j), cutval(s,i,j)*x(i,j)) =l= maxknight-1;
model knight1 / deftotal,defmovex,cut /;
s(ss) = no;
total.lo = maxknight - .5;
knight1.solvestat = %solvestat.NormalCompletion%;
knight1.modelstat = %modelstat.Optimal%;
loop(ss$(knight1.solvestat=%solvestat.NormalCompletion% and
(knight1.modelstat=%modelstat.Optimal% or
knight1.modelstat=%modelstat.IntegerSolution%)),
s(ss) = yes;
cutval(ss,i,j) = x.l(i,j);
solve knight1 maximizing total using mip; )
option cutval:0:0:1; display cutval;
```