Let’s do the Landau shuffle

Here’s a shuffle I’ve not seen before:

  1. Take an ordinary deck of 52 cards and place them, face up, in the following pattern:
    Going from the top of the deck to the bottom, placing cards down left-to-right, put 13 cards in the top row:
    \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\
    11 cards in the next row:
    \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\
    then 9 cards in the next row:
    \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\
    then 7 cards in the next row:
    \Box\ \Box\ \Box\ \Box\ \Box\ \Box\ \Box\
    then 5 cards in the next row:
    \Box\ \Box\ \Box\ \Box\ \Box\
    then 3 cards in the next row:
    \Box\ \Box\ \Box\
    and finally, the remaining 4 cards in the last row:
    \Box\ \Box\ \Box\ \Box\
  2. Now, take the left-most card in each row and move it to the right of the others (effectively, this is a cyclic shift of that row of cards to the left).
  3. Finally, reassemble the deck by reversing the order of the placement.

In memory of the great German mathematician Edmund Landau (1877-1938, see also this bio), I call this the Landau shuffle. As with any card shuffle, this shuffle permutes the original ordering of the cards. To restore the deck to it’s original ordering you must perform this shuffle exactly 180180 times. (By the way, this is called the order of the shuffle.) Yes, one hundred eighty thousand, one hundred and eighty times. Moreover, no other shuffle than this Landau shuffle will require more repetitions to restore the deck. So, in some sense, the Landau shuffle is the shuffle that most effectively rearranges the cards.

Now suppose we have a deck of n (distictly labeled) cards, where n>2 is an integer. The collection of all possible shuffles, or permutations, of this deck is denoted S_n and called the symmetric group. The above discussion leads naturally to the following question(s).

Question: What is the largest possible order of a shuffle of this deck (and how do you construct it)?

This requires a tiny bit of group theory. You only need to know that any permutation of n symbols (such as the card deck above) can be decomposed into a composition or product) of disjoint cycles. To compute the order of an element g \in S_n, write that element g in disjoint cycle notation. Denote the lengths of the disjoint cycles occurring in g as k_1, k_2, \dots , k_m, where k_i>0 are integers forming a partition of n: n = k_1 + k_2 +\dots + k_m. Then the order of g is known to be given by order(g) = LCM(k_1, k_2, ...., k_m), where LCM denotes the least common multiple.

The Landau function g(n) is the function that returns the maximum possible order of an element g \in S_n. Landau introduced this function in a 1903 paper where he also proved the asymptotic relation \lim_{n\to \infty} \frac{\ln(g(n))}{\sqrt{n\ln(n)}}=1.

Example: If n=52 then note 52 = 13+11+9+7+5+3+4 and that LCM(13,11,9,77,5,3,4)=180180.

Oddly, my favorite mathematical software program SageMath does not have an implementation of the Landau function, so we end with some SageMath code.

def landau_function(n):
L = Partitions(n).list()
lcms = [lcm(P) for P in L]
return max(lcms)

Here is an example (the time is included to show this took about 2 seconds on my rather old mac laptop):

sage: time landau_function(52)
CPU times: user 1.91 s, sys: 56.1 ms, total: 1.97 s
Wall time: 1.97 s

Coding Theory and Cryptography

This was once posted on my USNA webpage. Since I’ve retired, I’m going to repost it here.

Coding Theory and Cryptography:
From Enigma and Geheimschreiber to Quantum Theory

(David Joyner, ed.) Springer-Verlag, 2000.
ISBN 3-540-66336-3

Summary: These are the proceedings of the “Cryptoday” Conference on Coding Theory, Cryptography, and Number Theory held at the U. S. Naval Academy during October 25-26, 1998. This book concerns elementary and advanced aspects of coding theory and cryptography. The coding theory contributions deal mostly with algebraic coding theory. Some of these papers are expository, whereas others are the result of original research. The emphasis is on geometric Goppa codes, but there is also a paper on codes arising from combinatorial constructions. There are both, historical and mathematical papers on cryptography.
Several of the contributions on cryptography describe the work done by the British and their allies during World War II to crack the German and Japanese ciphers. Some mathematical aspects of the Enigma rotor machine and more recent research on quantum cryptography are described. Moreover, there are two papers concerned with the RSA cryptosystem and related number-theoretic issues.


  • Reminiscences and Reflections of a Codebreaker, by Peter Hilton pdf
  • FISH and I, by W. T. Tutte pdf
  • Sturgeon, The FISH BP Never Really Caught, by Frode Weierud, pdf
  • ENIGMA and PURPLE: How the Allies Broke German and Japanese Codes During the War, by David A. Hatch pdf
  • The Geheimschreiber Secret, by Lars Ulfving, Frode Weierud pdf
  • The RSA Public Key Cryptosystem, by William P. Wardlaw pdf
  • Number Theory and Cryptography (using Maple), by John Cosgrave pdf
  • A Talk on Quantum Cryptography or How Alice Outwits Eve, by Samuel J. Lomonaco, Jr. pdf
  • The Rigidity Theorems of Hamada and Ohmori, Revisited, by T. S. Michael pdf
  • Counting Prime Divisors on Elliptic Curves and Multiplication in Finite Fields, by M. Amin Shokrollahi pdf,
  • On Cyclic MDS-Codes, by M. Amin Shokrollahi pdf
  • Computing Roots of Polynomials over Function Fields of Curves, by Shuhong Gao, M. Amin Shokrollahi pdf
  • Remarks on codes from modular curves: MAPLE applications, by David Joyner and Salahoddin Shokranian, pdf

For more cryptologic history, see the National Cryptologic Museum.

This is now out of print and will not be reprinted (as far as I know). The above pdf files are posted by written permission. I thank Springer-Verlag for this.

Shanks’ SQUFOF according to McMath

In 2003, a math major named Steven McMath approached Fred Crabbe and I about directing his Trident thesis. (A Trident is like an honors thesis, but the student gets essentially the whole year to focus on writing the project.) After he graduated, I put a lot of his work online at the USNA website. Of course, I’ve retired since then and the materials were removed. This blog post is simply to try to repost a lot of the materials which arose as part of his thesis (which are, as official works of the US government, all in the public domain; of course, Dan Shanks notes are copyright of his estate and are posted for scholarly use only).

McMath planned to study Dan ShanksSQUFOF method, which he felt could be exploited more than had been so far up to that time (in 2004). This idea had a little bit of a personal connection for me. Dan Shanks was at the University of Maryland during the time (1981-1983) I was a grad student there. While he and I didn’t talk as much as I wish we had, I remember him as friendly guy, looking a lot like the picture below, with his door always open, happy to discuss mathematics.

One of the first things McMath did was to type up the handwritten notes we got (I think) from Hugh Williams and Duncan Buell. A quote from a letter from Buell:

Dan probably invented SQUFOF in 1975. He had just bought an HP 67 calculator, and the algorithm he invented had the advantages of being very simple to program, so it would fit on the calculator, very powerful as an algorithm when measured against the size of the program, and it factors double precision numbers using single precision arithmetic.

Here are some papers on this topic (with thanks to Michael Hortmann for the references to Gower’s papers):

David Joyner, Notes on indefinite integral binary quadratic forms with SageMath, preprint 2004. pdf

F. Crabbe, D. Joyner, S. McMath, Continued fractions and Parallel SQUFOF, 2004. arxiv

D. Shanks, Analysis and Improvement of the Continued Fraction Method of Factorization, handwritten notes 1975, typed into latex by McMath 2004. pdf.

D. Shanks, An Attempt to Factor N = 1002742628021, handwritten notes 1975, typed into latex by McMath 2004. pdf.

D. Shanks, SQUFOF notes, handwritten notes 1975, typed into latex by McMath 2004. pdf.

S. McMath, Daniel Shanks’ Square Forms Factorization, 2004 preprint. pdf1, pdf2.

S. McMath, Parallel Integer Factorization Using Quadratic Forms, Trident project, 2004. pdf.

Since that time, many excellent treatments of SQUFOF have been presented. For example, see

J. Gower, S. Wagstaff, Square Form Factorization, Math Comp, 2008. pdf.

J. Gower, Square Form Factorization, PhD Thesis, December 2004. pdf.

A mathematical card trick

If you search hard enough on the internet you’ll discover a pamphlet from the 1898 by Si Stebbins entitled “Card tricks and the way they are performed” (which I’ll denote by [S98] for simplicity). In it you’ll find the “Si Stebbins system” which he claims is entirely his own invention. I’m no magician, by from what I can dig up on Magicpedia, Si Stebbins’ real name is William Henry Coffrin (May 4 1867 — October 12 1950), born in Claremont New Hampshire. The system presented below was taught to Si by a Syrian magician named Selim Cid that Si sometimes worked with. However, this system below seems to have been known by Italian card magicians in the late 1500’s. In any case, this blog post is devoted to discussing parts of the pamphlet [S98] from the mathematical perspective.

In stacking the cards (face down) put the 6 of Hearts 6 \heartsuit first, the 9 of Spades 9 \spadesuit next (so it is below the 6 \heartsuit in the deck), and so on to the end, reading across left to right as indicted in the table below (BTW, the pamphlet [S98] uses the reversed ordering.) My guess is that with this ordering of the deck — spacing the cards 3 apart — it still looks random at first glance.

Hearts \heartsuitSpades \spadesuitDiamonds \diamondsuitClubs \clubsuit
Si Stebbins’ System

Next, I’ll present a more mathematical version of this system to illustrate it’s connections with group theory.

We follow the ordering suggested by the mnemonic CHaSeD, we identify the suits with numbers as follows: Clubs is 0, Hearts is 1, Spades is 2 and Diamonds is 3. Therefore, the suits may be identified with the additive group of integers (mod 4), namely: {\bf{Z}}/4{\bf{Z}} = \{ 0,1,2,3 \}.

For the ranks, identify King with 0, Ace with 1, 2 with 2, \dots, 10 with 10, Jack with 11, Queen with 12. Therefore, the ranks may be identified with the additive group of integers (mod 13), namely: {\bf{Z}}/13{\bf{Z}}=\{ 0,1,2,\dots 12\}.

Rearranging the columns slightly, we have the following table, equivalent to the one above.

Mathematical version of the Si Stebbins Stack

In this way, we identify the card deck with the abelian group

G = {\bf{Z}}/4{\bf{Z}} \times {\bf{Z}}/13{\bf{Z}} .

For example, if you spot the 2 \clubsuit then you know that 13 cards later (and If you reach the end of the deck, continue counting with the top card) is the 2 \heartsuit, 13 cards after that is the 2 \spadesuit, and 13 cards after that is the 2 \diamondsuit.

Here are some rules this system satisfies:

Rule 1 “Shuffling”: Never riff shuffle or mix the cards. Instead, split the deck in two, the “bottom half” as the left stack and the “top half” as the right stack. Take the left stack and place it on the right one. This has the effect of rotating the deck but preserving the ordering. Do this as many times as you like. Mathematically, each such cut adds an element of the group G to each element of the deck. Some people call this a “false shuffle” of “false cut.”

Rule 2 “Rank position”: The corresponding ranks of successive cards in the deck differs by 3.

Rule 3 “Suit position”: Every card of the same denomination is 13 cards apart and runs in the same order of suits as in the CHaSeD mnemonic, namely, Clubs \clubsuit, Hearts \heartsuit, Spades \spadesuit, Diamonds \diamondsuit.

At least, we can give a few simple card tricks based on this system.

Trick 1: A player picks a card from the deck, keeps it hidden from you. You name that card.

This trick can be set up in more than one way. For example, you can either

(a) spread the cards out behind the back in such a manner that when the card is drawn you can separate the deck at that point bringing the two parts in front of you, say a “top” and a “bottom” stack, or

(b) give the deck to the player, let them pick a card at random, which separates the deck into two stacks, say a “top” and a “bottom” stack, and have the player return the stacks separately.

You know that the card the player has drawn is the card following the bottom card of the top stack. If the card on the bottom of the top stack is denoted (a,\alpha) \in G and the card drawn is (b,\beta) then

b \equiv a+3 \pmod{13}, \ \ \ \ \ \ \beta \equiv \alpha +1 \pmod{4}.

For example, a player draws a card and you find that the bottom card is the 9 \diamondsuit. What is the card the player picked?

solution: Use the first congruence listed: add 3 to 9, which is 12 or the Queen. Use the second congruence listed: add one to Diamond \diamondsuit (which is 3) to get 0 \pmod 4 (which is Clubs \clubsuit). The card drawn is the Q \clubsuit.

Trick 2: Run through the deck of cards (face down) one at a time until the player tells you to stop. Name the card you were asked to stop on.

Place cards behind the back first taking notice what the bottom card is. To get the top card, add 3 to the rank of the bottom card, add 1 to the suit of the bottom card. As you run through the deck you silently say the name of the next card (adding 3 to the rank and 1 to the suit each time). Therefore, you know the card you are asked to stop on, as you are naming them to yourself as you go along.

A footnote to Robert H. Mountjoy

In an earlier post titled Mathematical romantic? I mentioned some papers I inherited of one of my mathematical hero’s Andre Weil with his signature. In fact, I was fortunate enough to go to dinner with him once in Princeton in the mid-to-late 1980s – a very gentle, charming person with a deep love of mathematics. I remember he said he missed his wife, Eveline, who passed away in 1986. (They were married in 1937.)

All this is simply to motivate the question, why did I get these papers? First, as mentioned in the post, I was given Larry Goldstein‘s old office and he either was kind enough to gift me his old preprints or left them to be thrown away by the next inhabitant of his office. BTW, if you haven’t heard of him, Larry was a student of Shimura, when became a Gibbs Fellow at Yale, then went to the University of Maryland at COllege Park in 1969. He wrote lots of papers (and books) on number theory, eventually becoming a full professor, but eventually settled into computers and data science work. He left the University of Maryland about the time I arrived in the early 1980s to create some computer companies that he ran.

This motivates the question: How did Larry get these papers of Weil? I think Larry inherited them from Mountjoy (who died before Larry arrived at UMCP, but more on him later). This motivates the question, who is Mountjoy and how did he get them?

I’ve done some digging around the internet and here’s what I discovered.

The earliest mention I could find is when he was listed as a recipient of an NSF Fellowship in “Annual Report of the National Science Foundation: 1950-1953” under Chicago, Illinois, Mathematics, 1953. So he was a grad student at the University of Chicago in 1953. Andre Weil was there at the time. (He left sometime in 1958.) Mountjoy could have gotten the notes of Andre Weil then. Just before Weil left Chicago, Walter Lewis Baily arrived (in 1957, to be exact). This is important because in May 1965 the Notices of the AMS reported that reported:

Mountjoy, Robert Harbison
Abelian varieties attached to representations of discontinuous groups (S. Mac Lane and W. L. Baily)

(His thesis was published posthumously in American Journal of Mathematics Vol. 89 (1967)149-224.) This thesis is in a field studied by Weil and Baily but not Saunders.

But we’re getting ahead of ourselves. The 1962 issue of Maryland Magazine had this:

Mathematics Grant
A team of University of Maryland mathematics researchers have received a grant of $53,000 from the National Science Foundation to continue some technical investigations they started two years ago.
The mathematical study they are directing is entitled “Problems in Geometric Function Theory.” The project is under the direction of Dr. James Hummel. Dr. Mischael Zedek. and Prof. Robert H. Mountjoy, all of the Mathematics Department. They are assisted by four graduate-student researchists. The $53,000 grant is a renewal of an original grant which was made two years ago.

We know he was working at UMCP in 1962. 

Here’s the sad news. 

The newspaper Democrat and Chronicle, from Rochester, New York, on Wednesday, May 25, 1965 (Page 40) published the news that Robert H. Mountjoy “Died suddenly at Purcellville, VA, May 23, 1965”. I couldn’t read the rest (it’s behind a paywall but I could see that much). The next day, they published more: “Robert H. Mountjoy, son-in-law of Mr and Mrs Allen P Mills of Brighton, was killed in a traffic crash in Virgina. Mountjoy, about 30, a mathematics instructors at the University of Maryland, leaves a widow Sarah Mills Mountjoy and a 5-month old son Alexander, and his parents Mr and Mrs Lucius Mountjoy of Chicago.”

It’s so sad. The saying goes “May his memory be a blessing.” I never met him, but from what I’ve learned of Mountjoy, his memory is indeed a blessing.

The number-theoretic side of J. Barkley Rosser

By chance, I ran across a reference to a paper of J Barkey Rosser and it brought back fond memories of days long ago when I would browse the stacks in the math dept library at the University of Washington in Seattle. I remember finding papers describing number-theoretic computations of Rosser and Schoenfeld. I knew nothing about Rosser. Was he a number theorist?


J. Barkley Rosser, taken at Math meeting in Denver

Here’s all I could glean from different sources on the internet:
J. Barkley Rosser was born in Jacksonville, Florida in 1907. He earned both his Bachelor of Science (1929) and his Master of Science (1931) from the University of Florida. Both degrees were in physics. He obtained his Ph.D. in mathematics (in fact, mathematical logic) from Princeton University in 1934, under the supervision of Alonso Church. After getting his Ph.D., Rosser taught at Princeton, Harvard, and Cornell and spent the latter part of his career at the University of Wisconsin-Madison. As a logician, Rosser is known for his part in the Church-Rosser Theorem and the Kleene–Rosser Paradox in lambda calculus. Moreover, he served as president of the Association for Symbolic Logic. As an applied mathematician, he served as president of the Society of Industrial and Applied Mathematics (otherwise known as SIAM). While at the University of Wisconsin-Madison, he served as the director of the U.S. Army Mathematics Research Center. He continued to lecture well into his late 70s, and passed away at his home in Madison in 1989. He has a son, J. Barkley Rosser Jr, who’s an economist at James Madison University.

What about Schoenfeld?


Lowell Schoenfeld spent his early years in New York City, graduating Cum Laude from the College of the City of New York in 1940. He went on to MIT to earn a Master’s. He received his Ph.D. in 1944 from the University of Pennsylvania under the direction of Hans Rademacher. (During his years in graduate school, he seems to have worked for the Philadelphia Navy Yard as well, writing reports on aircraft navigational computers.) After positions at Temple University and Harvard, he moved to the University of Illinois, where he met his future wife. He met Josephine M. Mitchell when she was a tenured Associate Professor and he was an untenured Assistant Professor. After they married, the University would no longer allow Mitchell to teach, so the couple both resigned their positions and eventually settled at Pennsylvania State University. They spent about 10 years there but in 1968 the couple moved to the University of Buffalo, where they remained until their retirements in the 1980s.

As far as I can tell, these are the papers they wrote together, all in analytic number theory:

[1] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximate formulas for some functions of prime numbers”. Illinois J. Math. 6 (1962), no. 1, 64–94.
[2] Rosser, J. Barkley; Schoenfeld, Lowell; J.M. Yohe. “Rigorous Computation and the Zeros of the Riemann Zeta-Function,” 1969
[3] Rosser, J. Barkley; Schoenfeld, Lowell. “Sharper Bounds for the Chebyshev Functions \theta (x) and \psi (x)” Mathematics of Computation Vol. 29, No. 129 (Jan., 1975), pp. 243-269
[4] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximation of the Riemann Zeta-Function” 1971.

I haven’t seen a copy of the papers [2] and [4] in years but I’m guessing these are what I looked at as a teenager in Seattle, years ago, wandering through the stacks at the UW.

Rosser also wrote papers on topics in recreational mathematics, such as magic squares. Two such papers were co-written with R.J. Walker from Cornell University, who’s more well-known for his textbook Algebraic Curves:

Rosser, Barkley; Walker, R. J. “The algebraic theory of diabolic magic squares,” Duke Math. J. 5 (1939), no. 4, 705–728
Rosser, Barkley; Walker, R. J. “On the transformation group for diabolic magic squares of order four,” Bull. Amer. Math. Soc. 44 (1938), no. 6, 416–420.

Diabolic magic squares, also called pan-diagonal magic squares, are n\times n squares of integers 1, 2, ..., n^2 whose rows all add to a constant C, whose columns all add to C, whose diagonals both add to C, and whose “broken diagonals” all add to C. An example was given by the German artist Albrecht Durer in the 1514 engraving called Melencolia I: (where C=34):


I wish I knew more about this number-theoretic side of Rosser. He’s a very  interesting mathematician.

Expected maximums and fun with Faulhaber’s formula

A recent Futility Closet post inspired this one. There, Greg Ross mentioned a 2020 paper by P Sullivan titled “Is the Last Banana Game Fair?” in Mathematics Teacher. (BTW, it’s behind a paywall and I haven’t seen that paper).

Suppose Alice and Bob don’t want to share a banana. They each have a fair 6-sided die to throw. To decide who gets the banana, each of them rolls their die. If the largest number rolled is a 1, 2, 3, or 4, then Alice wins the banana. If the largest number rolled is a 5 or 6, then Bob wins. This is the last banana game. In this post, I’m not going to discuss the last banana game specifically, but instead look at a related question.

Let’s define things more generally. Let I_n=\{1,2,...,n\}, let X,Y be two independent, uniform random variables taken from I_n, and let Z=max(X,Y). The last banana game concerns the case n=6. Here, I’m interested in investigating the question: What is E(Z)?

Computing this isn’t hard. By definition of independent and max, we have
P(Z\leq z)=P(X\leq z)P(Y\leq z).
Since P(X\leq z)=P(Y\leq z)={\frac{z}{n}}, we have
P(Z\leq z)={\frac{z^2}{n^2}}.
The expected value of Z is defined as \sum kP(Z=k), but there’s a handy-dandy formula we can use instead:
E(Z)=\sum_{k=0}^{n-1} P(Z>k)=\sum_{k=0}^{n-1}[1-P(Z\leq k)].
Now we use the previous computation to get
This solves the problem as stated. But this method generalizes in a straightforward way to selecting m independent r.v.s in I_n, so let’s keep going.

First, let’s pause for some background and history. Notice how, in the last step above, we needed to know the formula for the sum of the squares of the first n consecutive positive integers? When we generalize this to selecting m integers, we need to know the formula for the sum of the m-th powers of the first n consecutive positive integers. This leads to the following topic.

Faulhaber polynomials are, for this post (apparently the terminology is not standardized) the sequence of polynomials F_m(n) of degree m+1 in the variable n that gives the value of the sum of the m-th powers of the first n consecutive positive integers:

\sum_{k=1}^{n} k^m=F_m(n).

(It is not immediately obvious that they exist for all integers m\geq 1 but they do and Faulhaber’s results establish this existence.) These polynomials were discovered by (German) mathematician Johann Faulhaber in the early 1600s, over 400 years ago. He computed them for “small” values of m and also discovered a sort of recursive formula relating F_{2\ell +1}(n) to F_{2\ell}(n). It was about 100 years later, in the early 1700s, that (Swiss) mathematician Jacob Bernoulli, who referenced Faulhaber, gave an explicit formula for these polynomials in terms of the now-famous Bernoulli numbers. Incidentally, Bernoulli numbers were discovered independently around the same time by (Japanese) mathematician Seki Takakazu. Concerning the Faulhaber polys, we have
and in general,
F_m(n)={\frac{n^{m+1}}{m+1}}+{\frac{n^m}{2}}+ lower order terms.

With this background aside, we return to the main topic of this post. Let I_n=\{1,2,...,n\}, let X_1,X_2,...,x_m be m independent, uniform random variables taken from I_n, and let Z=max(X_1,X_2,...,X_m). Again we ask: What is E(Z)? The above computation in the m=2 case generalizes to:


For m fixed and n “sufficiently large”, we have


I find this to be an intuitively satisfying result. The max of a bunch of independently chosen integers taken from I_n should get closer and closer to n as (the fixed) m gets larger and larger.

Michael Reid’s Happy New Year Puzzles, 2018

Belatedly posted by permission of Michael Reid. Enjoy!

Here are some New Year’s puzzles to help start out 2018.

1. Arrange the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in the
expression a^b + c^d + e^f + g^h + i^j to make 2018.

2. (a) Express 2018 = p^q + r^s where p, q, r, s are primes.
(b) Express 2018 = p^q - r^s where p, q, r, s are primes.

3. (a) Is it possible to put the first 9 primes, 2, 3, 5, 7, 11, 13,
17, 19 and 23 into a 3×3 matrix that has determinant 2018?
(b) Is it possible to put the first 16 primes, 2, 3, 5, … , 53,
into a 4×4 matrix that has determinant 2018?

4. (a) Express 2018 = A / B using the fewest number of distinct
For example, the expression 7020622 / 3479 uses only seven
different digits. But it is possible to do better than this.
(b) Express 2018 = (A_1 \cdot A_2 \cdot ... \cdot A_m) / (B_1 \cdot B_2 \cdot ... \cdot B_n) using the fewest number of distinct digits.

Michael Reid’s Happy New Year Puzzle, 2017

Belatedly posted by permission of Michael Reid. Enjoy!

Here are some interesting puzzles to start the New Year; hopefully they are not too easy!

1. Express 2017 as a quotient of palindromes.

2. (a) Are there two positive integers whose sum is 2017 and whose product
is a palindrome?
(b) Are there two positive integers whose difference is 2017 and whose
product is a palindrome?

3. Is there a positive integer n such that both 2017 + n and
2017 n are palindromes?

4. What is the smallest possible sum of the decimal digits of 2017 n ,
where n is …
(a) … a positive integer?
(b) … a prime number?
(c) … a palindrome?

5. Consider the following two operations on a positive integer:
(i) replace a string of consecutive digits by its square,
(ii) if a string of consecutive digits is a perfect cube,
replace the string by its cube root.

Neither the string being replaced, nor its replacement, may have
have “leading zeros”. For example, from 31416 , we may change it to
319616 , by squaring the 14 . From 71253 , we may change it to
753 by taking the cube root of 125 .

(a) Starting from the number 2017 , what is the smallest number we
can reach with a sequence of these operations?
(b) What is the smallest number from which we can start, and reach
the number 2017 with a sequence of these operations?

6. Find a list of positive rational numbers, q_1 , q_2 , … , q_n
whose product is 1 , and whose sum is 2017 . Make your list as
short as possible.
Extra Credit: Prove that you have the shortest possible list.

Michael Reid’s Happy New Year Problems, 2020

Posted by permission of Michael Reid. Enjoy!

New Year’s Greetings!

Here are some fun puzzles to start the year.

1. Substitute the numbers 1, 2, … , 9 for the letters
a, b, … , i in the expression a^b + c^d + (e + f + g - h)^i
to get 2020.

2. Use the digits 1, 2, … , 9 in order, and any of the usual
arithmetic operations and parentheses to get a number that is
as close as possible to, but not exactly equal to 2020.

3. Express 2019/2020 as a sum of distinct Egyptian fractions,
i.e. 1 / n_1 + 1 / n_2 + ... + 1 / n_k for integers 0 < n_1 < n_2 < ...  n_k < 202049
(but 202049 is not square).

5. Make a 4×4 matrix of single-digit integers (0-9) with digits
2, 0, 2, 0 on the main diagonal, and having determinant 2020.
Is it possible to do it with a symmetric matrix?

If you liked this one, check out other puzzles ont this blog tagged with “Michael Reid”.