After finding the number of 3rd and 4th-order magic tori it was tempting to look further and Walter Trump kindly offered his computing skills to determine the number of the 5th-order magic tori.

Please note that the squares used to illustrate the magic tori are not necessarily "magic," except for the pandiagonal examples, as partially pandiagonal and basic magic tori display not only magic squares but also "semi-magic" cousins. To see the magic squares you sometimes need to move the viewpoint round the torus until a magic intersection of magic diagonals coincides with the centre of the grid, as explained in a previous article "From the Magic Square to the Magic Torus." With odd-order tori a magic intersection of magic diagonals occurs over a number and not over a space between numbers.

Provisionally, the different types of fifth-order tori are listed in descending magic order (taking into account the number of magic squares per torus). The results of Walter Trump's computing are thus as follows:

**T 5.01 Pandiagonal or Panmagic Tori Type 1**

with 10 crossed magic diagonals producing 25 magic intersectionswith 10 crossed magic diagonals producing 25 magic intersections

**Total : 144 pandiagonal or panmagic tori type 1 that display 3,600 pandiagonal or panmagic squares**

**T 5.02 Partially Pandiagonal Tori Type 2**

with 5 magic diagonals crossed by 3 other magic diagonals producing 15 magic intersectionswith 5 magic diagonals crossed by 3 other magic diagonals producing 15 magic intersections

**Total : 402 partially pandiagonal tori type 2 that display 6,030 partially pandiagonal squares and 4,020 semi-magic squares**

**T 5.03 Partially Pandiagonal Tori Type 3**

with 5 magic diagonals crossed by 2 other magic diagonals producing 10 magic intersectionswith 5 magic diagonals crossed by 2 other magic diagonals producing 10 magic intersections

**Total : 136 partially pandiagonal tori type 3 that display 1,360 partially pandiagonal squares and 2,040 semi-magic squares**

**T 5.04 Partially Pandiagonal Tori Type 4**

with 3 magic diagonals crossed by 3 other magic diagonals producing 9 magic intersectionswith 3 magic diagonals crossed by 3 other magic diagonals producing 9 magic intersections

**Total : 2,856 partially pandiagonal tori type 4 that display 25,704 partially pandiagonal squares and 45,696 semi-magic squares**

**T 5.05 Partially Pandiagonal Tori Type 5**

with 3 magic diagonals crossed by 2 other magic diagonals producing 6 magic intersectionswith 3 magic diagonals crossed by 2 other magic diagonals producing 6 magic intersections

**Total : 21,348 partially pandiagonal tori type 5 that display 128,088 partially pandiagonal squares and 405,612 semi-magic squares**

**T 5.06 Partially Pandiagonal Tori Type 6**

with 5 magic diagonals crossed by 1 other magic diagonal producing 5 magic intersectionswith 5 magic diagonals crossed by 1 other magic diagonal producing 5 magic intersections

**Total : 14,038 partially pandiagonal tori type 6 that display 70,190 partially pandiagonal squares and 280,760 semi-magic squares**

**T 5.07 Partially Pandiagonal Tori Type 7**

with 2 magic diagonals crossed by 2 other magic diagonals producing 4 magic intersectionswith 2 magic diagonals crossed by 2 other magic diagonals producing 4 magic intersections

**Total : 518,536 partially pandiagonal tori type 7 that display 2,074,144 partially pandiagonal squares and 10,889,256 semi-magic squares**

**T 5.08 Partially Pandiagonal Tori Type 8**

with 3 magic diagonals crossed by 1 other magic diagonal producing 3 magic intersectionswith 3 magic diagonals crossed by 1 other magic diagonal producing 3 magic intersections

**Total : 523,036 partially pandiagonal tori type 8 that display 1,569,108 partially pandiagonal squares and 11,506,792 semi-magic squares**

**T 5.09 Partially Pandiagonal Tori Type 9**

with 2 magic diagonals crossed by 1 other magic diagonal producing 2 magic intersectionswith 2 magic diagonals crossed by 1 other magic diagonal producing 2 magic intersections

**Total : 21,057,784 partially pandiagonal tori type 9 that display 42,115,568 partially pandiagonal squares and 484,329,032 semi-magic squares**

**T 5.10 Basic Magic Tori Type 10**

with 2 crossed magic diagonals producing 1 magic intersection (over a number) and one non-magic intersection (between numbers)with 2 crossed magic diagonals producing 1 magic intersection (over a number) and one non-magic intersection (between numbers)

**Total : 229,311,432 basic magic tori type 10 that display 229,311,432 basic magic squares and 5,503,474,368 semi-magic squares**

**GRAND TOTALS OF THE MAGIC TORI :**__251,449,712 FIFTH-ORDER MAGIC TORI THAT DISPLAY 275,305,224 MAGIC SQUARES____AND 6,010,937,576 SEMI-MAGIC SQUARES__

**T 5.11 Semi-Magic Tori Type 11**

with 5 parallel magic diagonals producing no magic intersectionswith 5 parallel magic diagonals producing no magic intersections

**Total : 64,108 semi-magic tori type 11 that display 1,602,700 semi-magic squares**

**T 5.12 Semi-Magic Tori Type 12**

with 3 parallel magic diagonals producing no magic intersectionswith 3 parallel magic diagonals producing no magic intersections

**Total : 3,823,164 semi-magic tori type 12 that display 95,579,100 semi-magic squares**

**T 5.13 Semi-Magic Tori Type 13**

with 2 parallel magic diagonals producing no magic intersectionswith 2 parallel magic diagonals producing no magic intersections

**Total : 187,679,364 semi-magic tori type 13 that display 4,691,984,100 semi-magic squares**

**T 5.14 Semi-Magic Tori Type 14**

with 1 magic diagonal producing no magic intersectionwith 1 magic diagonal producing no magic intersection

**Total : 4,138,786,720 semi-magic tori type 14 that display 103,469,668,000 semi-magic squares**

**T 5.15 Semi-Magic Tori Type 15**

with no magic diagonalswith no magic diagonals

**Total : 18,579,918,980 semi-magic tori type 15 that display 464,497,974,500 semi-magic squares**

**GRAND TOTALS OF THE SEMI-MAGIC TORI :**__22,910,272,336 FIFTH-ORDER SEMI-MAGIC TORI THAT DISPLAY__

__572,756,808,400 SEMI-MAGIC SQUARES__

**ALL TOGETHER:****23,161,722,048 FIFTH-ORDER MAGIC AND SEMI-MAGIC TORI THAT DISPLAY 275,305,224 MAGIC SQUARES**

AND 6,010,937,576 + 572,756,808,400 = 578,767,745,976 SEMI-MAGIC SQUARESAND 6,010,937,576 + 572,756,808,400 = 578,767,745,976 SEMI-MAGIC SQUARES

**THE TOTAL NUMBER OF FIFTH-ORDER MAGIC AND SEMI-MAGIC SQUARES IS**

579,043,051,200579,043,051,200

###
**CONCLUSIONS:**

**CONCLUSIONS:**

The study reveals the existence of 251,449,712 fifth-order magic tori and 22,910,272,336 fifth-order semi-magic tori.

The overall total of the fifth-order magic and semi-magic tori is 23,161,722,048 which is equal to the number of fifth-order magic and semi-magic squares (see Walter Trump's previous enumeration) divided by 25, (5² being the number of squares per torus for 5

251,449,712 is now the fifth number of the sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

23,161,722,048 (251,449,712 + 22,910,272,336) is now the fifth number of the sequence A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

^{th}-order tori).251,449,712 is now the fifth number of the sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

23,161,722,048 (251,449,712 + 22,910,272,336) is now the fifth number of the sequence A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS).

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