*Melencolia I, a 1514 engraving by Albrecht Dürer (1471–1528), now located at the Staatliche Kunsthalle Karlsruhe. This engraving contains a 4x4 magic square based on the number 34. The sum 34 is found in the rows, columns, diagonals, but also in each of the quadrants and the center four squares, among other places. (Via Wikimedia Commons.)*

Palindromes are words or sentences that read the same forwards and backwards. A good technological example of a palindrome is the word,

This gives us Gauss' answer when

*Carl Friedrich Gauss (1777-1855).Gauss has a plethora of things named after him, including a unit of magnetism, the gauss, that's been superseded by the tesla.As I wrote in a recent article (Great Circle Routes, June 25, 2018), Gauss did an experiment in the 1820s to determine whether space was curved; that is, non-Euclidean.He used his era's version of a laser theodolite, called a heliotrope, to survey a triangle between three mountains - Brocken, Hohenhagen, and Inselberg.This triangle had sides of length 69, 85 and 107 kilometers, but even such a large triangle has the now known difference in the sum of angles from 180 degrees of just 0.1 picoradians.(An 1840 oil portrait by the danish painter Christian Albrecht Jensen (1792-1870), via Wikimedia Commons)*

If we write the decimal digits 1-9 in sequence and insert a random addition or multiplication operation between digits, as in

This serves as an introduction to a lengthy 2014 paper on arXiv by Inder J. Taneja, who was formerly a professor of mathematics at the Universidade Federal de Santa Catarina (Florianópolis, Brazil).[2] Taneja considers both ascending and descending decimal digits, parenthetical grouping, and also adds exponentiation to the mathematical operator mix. Also allowed are numbers formed by the concatenation of successive digits, as the following examples show.1 x 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 44

1 + 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362881

By adding exponentiation, we can create numbers less than 44, as the example,346 = 1^{2345}+ 6 x 7 x 8 + 9

346 = 9 x 8 + 7 + 65 x 4 + 3 x 2 + 1

514 = 1 + 2^{3}x 4 + 56 x 7 + 89

514 = 98 + 76 x 5 + 4 + 32 x 1

6096 = 1^{2}x 3 x 4^{5}+ 6 x 7 x 8 x 9

6096 = 9 + 87 x 65 + 432 x 1

9261 = 12^{3}x 4 + 5 x 6 x 78 + 9

9261 = (9 + 8 + 76 + 54) x 3 x 21

*Old school numerical display - An array of IN16 Nixie tubes. The instruments in my early laboratories contained mostly analog voltmeters, often with a mirrored strip to allow an accurate reading of the indicator position without parallax. The first digital instruments had displays using these Nixie tubes. (Created from a Wikimedia Commons image by "FxDev.")*

- Clever Carl, NRICH team of the Millennium Mathematics Project, February, 2011.
- Inder J. Taneja, "Crazy Sequential Representation: Numbers from 1 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9," arXiv, January 8, 2014.
- Tim Wylie, "Crazy Sequential Representations of Numbers for Small Bases," arXiv, October 11, 2018.