After studying the puzzle I believe that finding a solution by hand would be feasible for a good mathematician with time to spare. I also believe that the number of possible solutions is quite big.
Using a computer program I have found a solution:
-42 22 23 7
13 11 -32 14
-23 16 15 8
19 9 -22 1
It has the following square sums:
4 24 12 3 84 39
17 10 5 6 20
21 18 2
My program use simulated annealing, which is somewhat unconventional for a puzzle that asks for an exact solution.
The trick is to find a method of measuring the quality of a solution candidate, along with a method of transforming it into "neighbours" that fit both one another and the concept of simulated annealing in general.
My program measure quality by first sorting the list of all 30 square values, then for each of the 7 sub-lists of length 24 it compares each number to the corresponding solution number, and add the absolute difference to the score for the sub-list, the lowest score of these 7 sub-lists is then the score for the solution candidate, and the lower a candidate scores the closer it is considered to be to a solution.
In each iteration step my program will create new candidates by picking a random selection of the 16 numbers, possibly picking some of them multiple times, and for each pick randomly add either -1, 0 or 1. Thus keeping the change of the quality measure bounded by the number of picks.
My code use numbers 500 greater than the actual solution numbers in order to make sorting a bit simpler, that is just a JavaScript thing. I also skip checking the two first of the 7 sub-lists as I have proven that there must be at least 2 negative numbers in a solution. It is quickly written code, but here goes:
<!DOCTYPE HTML>
<html>
<head>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<title>Integer Grid</title>
</head>
<body>
<script>
var grid=[]
var a,b,c,d
for(a=0;a<30;a++){
grid[a]=500
}
var scramble
var iterations
var newgrid
var sortgrid
var score
var minscore
var branches
var best
var bestscore
for(scramble=50;scramble>0;scramble--){
for(iterations=0;iterations<10000;iterations++){
bestscore=1000000
for(branches=0;branches<30;branches++){
newgrid=grid.slice(0)
for(a=0;a<scramble;a++){
b=Math.floor(Math.random()*16)
newgrid[b]+=Math.floor(Math.random()*3)-1
}
newgrid[16]=newgrid[0]+newgrid[1]+newgrid[4]+newgrid[5]-1500
newgrid[17]=newgrid[2]+newgrid[1]+newgrid[6]+newgrid[5]-1500
newgrid[18]=newgrid[2]+newgrid[3]+newgrid[6]+newgrid[7]-1500
newgrid[19]=newgrid[8]+newgrid[9]+newgrid[4]+newgrid[5]-1500
newgrid[20]=newgrid[10]+newgrid[9]+newgrid[6]+newgrid[5]-1500
newgrid[21]=newgrid[10]+newgrid[11]+newgrid[6]+newgrid[7]-1500
newgrid[22]=newgrid[8]+newgrid[9]+newgrid[12]+newgrid[13]-1500
newgrid[23]=newgrid[10]+newgrid[9]+newgrid[14]+newgrid[13]-1500
newgrid[24]=newgrid[10]+newgrid[11]+newgrid[14]+newgrid[15]-1500
newgrid[25]=newgrid[16]+newgrid[20]+newgrid[2]-newgrid[5]+newgrid[8]-1000
newgrid[26]=newgrid[18]+newgrid[20]+newgrid[1]-newgrid[6]+newgrid[11]-1000
newgrid[27]=newgrid[22]+newgrid[20]+newgrid[4]-newgrid[9]+newgrid[14]-1000
newgrid[28]=newgrid[24]+newgrid[20]+newgrid[7]-newgrid[10]+newgrid[13]-1000
newgrid[29]=newgrid[16]+newgrid[18]+newgrid[22]+newgrid[24]-1500
sortgrid=newgrid.slice(0).sort()
minscore=1000000
for(a=2;a<=6;a++){
score=0
b=a+24
for(c=a,d=501;c<b;c++,d++){
score+=Math.abs(d-sortgrid[c])
}
minscore=Math.min(minscore,score)
}
if(minscore<bestscore){
bestscore=minscore
best=newgrid
}
}
grid=best
if(bestscore===0){
console.log("Solution:")
console.log(grid)
throw new Error("Done")
}
}
console.log(grid)
}
</script>
</body>
</html>
0 to 25
I have experimented a bit with extending the puzzle, the best I have is a solution with all the numbers from 0 to 25, but 1 to 26 have eluded me, maybe it is impossible, maybe I just haven't hit it. I expect that it is in general easier to find solutions where the average of the series is closer to 0.
16 15 8 0 19 1 23 25 18 87
13 -25 3 12 4 -34 11 14 22
7 9 -21 17 24 10 21
6 2 20 5
-7 to 21
30 consecutive numbers is impossible, as the sum of 30 consecutive numbers is always odd, and the sum of the 30 numbers in the puzzle is always even. But 29 consecutive is doable, the highest such series I have found run from -7 to 21:
9 -4 12 13 15 10 18 17 16 55
6 4 -2 -5 0 1 -6 20 21
-7 -3 2 -1 3 14 19
5 8 7 11
Extreme solutions
Inspired by Florian F's observations about big and small numbers I decided to find some solutions that include really high and really low values, the best i have managed to find is a solution that include the number -99, and one that include 63 in the 16 base numbers:
-99 11 4 13 -60 23 24 -45 16 1
22 6 2 5 17 3 7 77 15
14 -25 20 -20 18 12 19
21 8 9 10
63 3 1 13 15 20 23 22 9 73
-62 11 5 4 -68 7 8 2 24
10 -27 18 -19 16 17 19
21 12 14 6
