MULTIMAGIE.COM - Magic squares of cubes: order 12

12x12 magic squares of cubes.
12x12 magic squares of 4th powers.
12x12 magic squares of 5th powers.
See also the Magic squares of cubes general page

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) a^N + b^N + c^N = d^N + e^N + f^N = g^N + h^N + i^N = j^N + k^N + l^N = y
(12.2) m^N + n^N + o^N + p^N = q^N + r^N + s^N + t^N = u^N + v^N + w^N + x^N = z

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

*François Labelle's method to construct 12x12 semi-magic squares of Nth-powers*
(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:

April 2010: first known 12x12 semi-magic square of 4th powers
by François Labelle, S4 = 53809938 * 650658 = 35011866639204
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:

March 2018: 12x12 semi-magic square of consecutive 4th powers,
with one magic diagonal, by Nicolas Rouanet, S4 = 1014123110
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.

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