if your friend has sufficiently mathematical knowledge there are ways to convince your friend by using a computer program :-)
Some time ago i saw a similiar problem where the numbers were percentages https://groups.google.com/group/de.rec.denksport/browse_thread/thread/aa7dde6a6043d7c4.
So the solution does not have to be an integer. Then one can formulate the problem as linear programming problem. The meaning of the variable $x1110$ for example is: it is the percentage of soldiers that have the first property "lost a left leg" (first index of variable $x$ is
$1$), the second property "lost a right leg" (second index of variable $x$ is $1$), the third property "lost a left arm"
(third index of variable $x$ is $1$) but not the fourth property "lost a right arm" (fourth index of variable $x$ is $0$). The four statements paritions the set of all soldiers in $2^4$ subsets and the following holds:
the percentages are between 0 and 100 (c1 ... c32) , the percentages of soldiers with the first,...,fourth property is 70,75,80,85 (c33,...,c34) and the sum of all percentages is 100. we are interested in the case where the number of soldiers that have lost all four limbs is minimal (min). We can use this input directly in the online linear optimization solver I found and get the expected solution. If all calculation are done in exact arithmetic (I think the applet does not) then 10 is the minimal integer solution.
var x0000>=0;
var x0001>=0;
var x0010>=0;
var x0011>=0;
var x0100>=0;
var x0101>=0;
var x0110>=0;
var x0111>=0;
var x1000>=0;
var x1001>=0;
var x1010>=0;
var x1011>=0;
var x1100>=0;
var x1101>=0;
var x1110>=0;
var x1111>=0;
minimize z: x1111;
subject to c17: x0000 <= 100;
subject to c18: x0001<=100;
subject to c19: x0010<=100;
subject to c20: x0011<=100;
subject to c21: x0100<=100;
subject to c22: x0101<=100;
subject to c23: x0110<=100;
subject to c24: x0111<=100;
subject to c25: x1000<=100;
subject to c26: x1001<=100;
subject to c27: x1010<=100;
subject to c28: x1011<=100;
subject to c29: x1100<=100;
subject to c30: x1101<=100;
subject to c31: x1110<=100;
subject to c32: x1111<=100;
subject to c33: x1000 + x1001 + x1010 + x1011 + x1100 +x1101 + x1110 +x1111 = 70;
subject to c34: x0100 + x0101 + x0110 + x0111 + x1100 +x1101 + x1110 +x1111 = 75;
subject to c35: x0010 + x0011 + x0110 + x0111 + x1010 +x1011 + x1110 +x1111 = 80;
subject to c36: x0001 + x0011 + x0101 + x0111 + x1001 +x1011 + x1101 +x1111 = 85;
subject to c37: x0000 + x0001 + x0010 +x0011 + x0100 + x0101 + x0110 +x0111 + x1000 + x1001 + x1010 +x1011 + x1100 + x1101 + x1110 + x1111 = 100;
end;
solution:
x1111 = 10.0
x0111 = 30.0
x1011 = 25.0
x1101 = 20.0
x1110 = 15.0
all other variables (percentages) are 0