12x12 magic squares of cubes.
12x12 magic squares of 4th powers.
12x12 magic squares of 5th powers.

At least one 12x12 magic square of cubes is known: using the 12x12 trimagic square of Walter Trump constructed in 2002, it is easy to obtain a magic square of cubes, directly raising to the third power its integers.

François Labelle, inspired by Lee Morgenstern's 6x6 method, found in April 2010 a method to construct 12x12 semi-magic squares of fourth powers (or of any Nth-powers).

If the two equations (12.1) (12.2) are true:

• (12.1)    aN + bN + cN = dN + eN + fN = gN + hN + iN = jN + kN + lN = y
• (12.2)    mN + nN + oN + pN = qN + rN + sN + tN = uN + vN + wN + xN = z

then this square is a semi-magic square of Nth-powers, with magic sum SN = yz:

 (am)N (an)N (ao)N (ap)N (bq)N (br)N (bs)N (bt)N (cu)N (cv)N (cw)N (cx)N (bm)N (bn)N (bo)N (bp)N (cq)N (cr)N (cs)N (ct)N (au)N (av)N (aw)N (ax)N (cm)N (cn)N (co)N (cp)N (aq)N (ar)N (as)N (at)N (bu)N (bv)N (bw)N (bx)N (dn)N (do)N (dp)N (dm)N (er)N (es)N (et)N (eq)N (fv)N (fw)N (fx)N (fu)N (en)N (eo)N (ep)N (em)N (fr)N (fs)N (ft)N (fq)N (dv)N (dw)N (dx)N (du)N (fn)N (fo)N (fp)N (fm)N (dr)N (ds)N (dt)N (dq)N (ev)N (ew)N (ex)N (eu)N (go)N (gp)N (gm)N (gn)N (hs)N (ht)N (hq)N (hr)N (iw)N (ix)N (iu)N (iv)N (ho)N (hp)N (hm)N (hn)N (is)N (it)N (iq)N (ir)N (gw)N (gx)N (gu)N (gv)N (io)N (ip)N (im)N (in)N (gs)N (gt)N (gq)N (gr)N (hw)N (hx)N (hu)N (hv)N (jp)N (jm)N (jn)N (jo)N (kt)N (kq)N (kr)N (ks)N (lx)N (lu)N (lv)N (lw)N (kp)N (km)N (kn)N (ko)N (lt)N (lq)N (lr)N (ls)N (jx)N (ju)N (jv)N (jw)N (lp)N (lm)N (ln)N (lo)N (jt)N (jq)N (jr)N (js)N (kx)N (ku)N (kv)N (kw)N

Here is his solution of fourth powers, giving the smallest magic sum and 144 distinct integers:

• 24 + 714 + 734 = 174 + 624 + 794 = 294 + 534 + 824 = 374 + 464 + 834 = 53809938
• 24 + 164 + 214 + 254 = 54 + 114 + 124 + 284 = 134 + 194 + 204 + 244 = 650658

generating the square:

 44 324 424 504 3554 7814 8524 19884 9494 13874 14604 17524 1424 11364 14914 17754 3654 8034 8764 20444 264 384 404 484 1464 11684 15334 18254 104 224 244 564 9234 13494 14204 17044 2724 3574 4254 344 6824 7444 17364 3104 15014 15804 18964 10274 9924 13024 15504 1244 8694 9484 22124 3954 3234 3404 4084 2214 12644 16594 19754 1584 1874 2044 4764 854 11784 12404 14884 8064 6094 7254 584 4644 6364 14844 2654 5834 16404 19684 10664 15584 11134 13254 1064 8484 9844 22964 4104 9024 5804 6964 3774 5514 17224 20504 1644 13124 3484 8124 1454 3194 10604 12724 6894 10074 9254 744 5924 7774 12884 2304 5064 5524 19924 10794 15774 16604 11504 924 7364 9664 23244 4154 9134 9964 8884 4814 7034 7404 20754 1664 13284 17434 10364 1854 4074 4444 11044 5984 8744 9204

This method can't be applied to 5th powers: because nobody knows a Taxicab(5, 3, 3) number (means a5 + b5 + c5 = d5 + e5 + f5 = g5 + h5 + i5), it will be very difficult to find a solution of the more difficult (12.1) equation which is a Taxicab(5, 3, 4) number!

Open problems:

• 12x12 magic squares of 4th powers are unknown.
• 12x12 magic squares of 5th powers are unknown.

March 2018, Nicolas Rouanet, France, constructed this 12x12 (and also a 10x10) nearly-magic square of consecutive 4th powers from 0^4 to 143^4:

 444 624 1304 514 1154 384 684 94 1274 1234 164 634 1424 574 394 344 24 124 1394 54 1134 504 844 314 154 1004 554 1434 214 1204 674 584 904 104 1094 804 1164 694 204 834 1024 34 1224 994 524 614 924 1254 284 1354 794 484 1044 734 534 1344 604 714 1014 664 1294 894 304 474 1084 774 754 814 64 1384 884 224 04 874 184 1414 174 724 1054 744 364 984 974 1194 294 644 704 654 1064 1364 244 1244 1074 1034 494 234 324 194 1404 424 354 1214 864 264 594 414 1314 824 934 544 954 764 784 134 564 374 1374 1144 854 1184 14 964 1174 404 1334 254 1114 1284 464 84 144 334 914 1324 114 944 74 1124 44 454 434 274 1104 1264

Nicolas Rouanet remarked, using a reasoning modulo 5, that a 12x12 magic (or semi-magic) square is impossible using consecutive 4th powers from 1^4 to 144^4.