4 4 7 2
8 3 7 8
4 1 7 5
6 4 4 9
The first step is to determine the possible values of P, Q, and R. I used the following javascript:
for (var p=10; p<100; p++) {
for (var q=10; q<100; q++) {
for (var r=10; r<100; r++) {
var total=Math.pow(p,3)+Math.pow(q,3)+Math.pow(r,3);
var result=p*10000+q*100+r;
if (total==result) {
document.write(p + "³ + ");
document.write(q + "³ + ");
document.write(r + "³ = ");
document.write(result + "<br>");
}
}
}
}
to come up with:
16³ + 50³ + 33³ = 165033
22³ + 18³ + 59³ = 221859
34³ + 10³ + 67³ = 341067
44³ + 46³ + 64³ = 444664
48³ + 72³ + 15³ = 487215
98³ + 28³ + 27³ = 982827
98³ + 32³ + 21³ = 983221
Then, looking at the table, we see that two of the P-values are a and a/2. These, therefore, must be 44 and 22. The first two rows, then, must correspond to 44³+46³+64³=444664 and 22³+18³+59³=221859, respectively. That gives us:
a = 44 g =
b = h =
c = 72 j = 64
d = k = 46
e = m =
f = n = 59
4 4 7 2
· · · ·
4 · · 5
6 4 · 9
From the determined values of a and j, the fourth row corresponds to 16³+50³+33³=165033, and g=175:
a = 44 g = 175
b = h =
c = 72 j = 64
d = k = 46
e = m =
f = n = 59
4 4 7 2
· · · ·
4 1 7 5
6 4 · 9
Also, we see that 2m is used twice as a P-value. Therefore, 2m must equal 98, and m=49. Also, the last row has Q=j/2, which we already know is 32 (j=64). Therefore, the last two rows are 98³+28³+27³=982827 and 98³+32³+21³=983221, respectively, and we are able to determine d, e, and f:
a = 44 g =
b = h =
c = 72 j = 64
d = k = 46
e = m =
f = n = 59
4 4 7 2
· · · ·
4 · · 5
6 4 · 9
From the determined values of a and j, the fourth row corresponds to 16³+50³+33³=165033, and g=175:
a = 44 g = 175
b = h =
c = 72 j = 64
d = 777 k = 46
e = 837 m = 49
f = 28 n = 59
4 4 7 2
8 3 7 8
4 1 7 5
6 4 4 9
From the completed squares in the grid, now, we have all the values. They are:
a = 44 g = 175
b = 48 h = 314
c = 72 j = 64
d = 777 k = 46
e = 837 m = 49
f = 28 n = 59
The sets of (P, Q, R) in the table are, in order:
(44, 46, 64)
(22, 18, 59)
(48, 72, 15)
(16, 50, 33)
(98, 28, 27)
(98, 32, 21)
Original puzzle by John Gowland |
