Using your Head is Permitted

April 2010 riddle

This month's riddle was suggested by David Jager.

Define a "Self-Assembled Number" (SAN) to be a positive integer that is the result of a calculation involving its own digits plus the four basic arithmetic operations and exponentiation, where each digit is used exactly once. Notably, the definition of SAN is dependent on the base in which the integer is represented. Here are some examples of SANs:

In base ten:
(i) 25 = 52
(ii) 625 = 56-2
(iii) 1024 = (4/2)10

In binary:
(iv) 11001TWO = 101TWO10TWO

(Interestingly, examples (i) and (iv) are really the same calculation in a different representation base.)

Note that, in the calculation, digits can also be concatenated into integers, such as in the case of the "10" in examples (iii) and (iv). We do not allow, however, extraneous leading zeros, decimal points and the like.

This month's riddle has two parts. Answer either or both to be credited as a solver. There will be separate solvers' lists kept for each part.

In each of the parts, you are asked to determine about a specific subset of SANs in which bases there is an infinite number of them. For any base of representation where you claim an infinite number does not exist, prove this point. Where you claim it does exist, show a construction.

Part 1:

Consider only SANs where the calculation uses the digits in the same order as they appear in the resulting number. For example:

127 = -1 + 27

In which bases are there an infinite number of these?

Part 2:

Consider only SANs where the calculation is the multiplication of exactly two operands, both having the same length in digits. In which bases are there an infinite number of these?

In order to be considered a solver for Part 2, you should present an answer for all bases that are composite numbers. The answer for primes is a bonus question.

(The published answer will cover both cases.)

As another bonus question, consider the numbers described in Part 1. In which bases are there an infinite number of these that are also prime numbers?

The solution to this month's riddle will include references to further reading. Until these references are published, readers are kindly asked to refrain from Internet searching on the topic if they want to qualify as original solvers.

One last word: "Using your Head is Permitted" and IBM's "Ponder This" decided to coordinate themes this month on their respective riddles. As a result, readers interested in more riddles on a similar vein can also refer to the Ponder This site.

List of solvers:

Part 1:

Jin Ruizhang (3 April 05:53)
Dan Dima (3 April 15:37)
Daniel Bitin (6 April 22:51)
Yongxing Deng (8 April 05:07) including the related bonus question!
Joseph DeVincentis (8 April 18:24)
Bojan Bašić (18 April 07:37) including the related bonus question!
Ganesh Lakshminarayana (18 April 23:13)
Dan Colestock (21 April 03:17)
Phil Muhm (28 April 02:29)

Part 2:

Dan Dima (2 April 21:43)
Jin Ruizhang (3 April 07:56)
Daniel Bitin (6 April 22:51)
Joseph DeVincentis (9 April 16:00)
Yongxing Deng (12 April 04:33) including the related bonus question!
Bojan Bašić (18 April 07:37) including the related bonus question!
Phil Muhm (23 April 00:24)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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