Power Constellations

Here is an example of a single constellation.

2 + 3 + 7 = 1 + 5 + 6

(i) Square

It can be written as a square constellation.

2 + 3 + 7 = 1 + 5 + 6

22 + 32 + 72 = 12 + 52 + 62

(ii) Cube

Here some even more remarkable constellations that are collectively single, squared and cubed.

0 + 5 + 5 + 10 = 1 + 2 + 8 + 9                   Single

02 + 52 + 52 + 102 = 12 + 22 + 82 + 92         Squared

03 + 53 + 53 + 103 = 13 + 23 + 83 + 93           Cubed

1 + 4 + 12 + 13 + 20 = 2 + 3 + 10 + 16 + 19                       Single

12 + 42 + 122 + 132 + 202 = 22 + 32 + 102 + 162 + 192             Squared

13 + 43 + 123 + 133 + 203 = 23 + 33 + 103 + 163 + 193             Cubed

(iii) Quintic

Here is one very impressive constellation containing 5 stages.

1 + 6 + 7 + 17 + 18 + 23 = 2 + 3 + 11 + 13 + 21 + 22

12 + 62 + 72 + 172 + 182 + 232 = 22 + 32 + 112 + 132 + 212 + 222

13 + 63 + 73 + 173 + 183 + 233 = 23 + 33 + 113 + 133 + 213 + 223

14 + 64 + 74 + 174 + 184 + 234 = 24 + 34 + 114 + 134 + 214 + 224

15 + 65 + 75 + 175 + 185 + 235 = 25 + 35 + 115 + 135 + 215 + 225

The Moscow Puzzles (Boris Kordemsky 1975)

Reversed Constellations

(i) Square

Here is a truly amazing power constellation.

Consider the following square constellation.

42 + 52 + 62 = 22 + 32 + 82

If we place each of the digits on the right hand side to the back of their corresponding digits on the left hand side, we have:

422 + 532 + 682

Then if we place each of the digits on the left hand side to the back of their corresponding digits on the right hand side, we have:

242 + 352 + 862

Remarkably, these two expressions equal each other!

422 + 532 + 682 = 242 + 352 + 862

Incredibly, it turns out that there are further possible combinations, that is, the transfer of the digits needn’t be to their corresponding digits.

There are 5 more combinations. See if you can identify them.

Click to see solution..

Solution

422 + 582 + 632 = 242 + 362 + 852

432 + 522 + 682 = 252 + 342 + 862

432 + 582 + 622 = 262 + 342 + 852

482 + 522 + 632 = 252 + 362 + 842

482 + 532 + 622 = 262 + 352 + 842

(ii) Cube

Consider the following reversed constellation of twelve numbers that can be reversed single, squared and cubed.

13 + 42 + 53 + 57 + 68 + 97 = 31 + 24 + 35 + 75 + 86 + 79                       Single

132 + 422 + 532 + 572 + 682 + 972 = 312 + 242 + 352 + 752 + 862 + 792       Squared

133 + 423 + 533 + 573 + 683 + 973 = 313 + 243 + 353 + 753 + 863 + 793        Cubed

There is another combination of these digits, which produces another reversed Constellation. See if you can find it.

Click to see solution..

Solution

12 + 32 + 43 + 56 + 67 + 87 = 21 + 23 + 34 + 65 + 76 + 78                       Single

122 + 322 + 432 + 562 + 672 + 872 = 212 + 232 + 342 + 652 + 762 + 782       Squared

123 + 323 + 433 + 563 + 673 + 873 = 213 + 233 + 343 + 653 + 763 + 783        Cubed

The Moscow Puzzles (Boris Kordemsky 1975)