The Journal of Recreational Mathematics (JRM) has now ceased publication, but here are some of my contributions to, or comments on, some of the contents of the Problems and Conjectures section of that journal. After 2015, it was followed by the Topics in Recreational Mathematics (TRM) edited by Charles Ashbacher and available from Amazon.
The questions are simply listed here with links to the individual documents where my comments and solutions are to be found, together with detailed references in each case to the relevant issue of the journal. All the links are to PDF files, except where otherwise indicated.
I’ve edited some for clarity or convenience, and added some extra comments where that seemed appropriate. My extra comments are usually enclosed in []. Sometimes I found time to extend the results further. I also fixed any editorial errors I found, but essentially these contributions are as I submitted them, if I did. Not all were published, and I have selected only those where the contribution appeared to add to the solution provided. The large gaps in the sequence is due to the fact that there were periods when I had no time to look at them to try to find solutions. And, of course, some simply did not interest me enough.
Question 2534 in JRM 30(3). Answer in JRM 31(3).
Show that the thirteenth day of each month falls on a Friday more often than any other day of the week.
Question 2536 in JRM 30(3). Answer in JRM 31(3)
Four people of different ages have each told each other how old they are.
One of them then said “If I multiply my age by any of your ages, the product
is a permutation of the digits in the two ages.”
How old is everyone?
Question 2537 in JRM 30(3). Answer in JRM 31(3)
The following description of Morra: A Game for Two appears in Math Puzzles and Games by M. Holt, Dorset Press, 1992.
This very old finger game comes from Italy. One player is called Morra. On a given signal — a nod of the head, for example — both players put up either one or two fingers, both at the same time. The payoff rules are, in summary form:
In his analysis, Holt displays the payoff matrix, then simply states “Morra’s best strategy, to reduce his losses, is to show two fingers all the time, then he never loses more than one penny.”
Can you find a better strategy, which causes Morra’s expected losses below one penny per game, or if possible lets him make an expected profit?Question 2538 in JRM 30(3). Answer in JRM 31(3)
Determine all ways to dissect a square into three similar parts, two of which are
congruent.
Editor’s note: Similar allows mirror images.
Question 2541 in JRM 30(3). Answer in JRM 31(3)
This problem was brought to the attention of the diarist Samuel Pepys in 1693 by a “Mr. Smith”. Pepys knew a number of eminent scientists and mathematicians through his connections with the Royal Society, so on November 22, 1693, he sent with a letter of introduction to Isaac Newton. Newton found the question badly worded in its original form, but Smith was able to clear up the ambiguities. Three weeks later, Newton wrote back to Pepys giving the correct solution and some details of his working
The problem concerns three men, A, B, and C, playing the following dice game:
Who has the best chance of winning? More specifically:
Question 2543 in JRM 30(3). Answer in JRM 31(3)
George, his wife Georgina, and their daughter Georgette have joined the Pandigital Society. Each member has a Membership Number and a Secret Code Number used for encrypting messages. The Secret Code is always a pandigital number, and is always the square of the Membership Number, which is not very secret, but that’s the way it is.
The society defines a pandigital number as one containing all ten digits once each. Neither Membership Numbers nor Secret Codes may have leading zeros.
On checking their Membership Cards, Georgette discovered that her Membership Number contains all the digits which do not appear in either of her parents’ Membership Numbers.
I can tell you that the difference between George’s Secret Code and Georgina’s is 2048627457 — can you tell me Georgette’s Membership Number?
Question 2544 in JRM 30(3). Answer in JRM 31(3).
I found a better solution, and proved it minimal, which was published in JRM 35(1)
Dennis Shasha featured this interesting form of puzzle in Scientific American September 2001. My puzzle, unlike his, really is square!
A square dance floor is marked out on an 8 x 8 grid of squares. Each square is occupied by a dancer. Initially, thirtytwo women (indicated by white circles in the first diagram) surround thirtytwo men (black circles). The choreographer wants them to change places (creating the second diagram) using a minimum of steps.
At each step, all the dancers move simultaneously — each dancer may either “step on the spot” remaining on the same square, or step into an orthogonally adjacent square. Two adjacent dancers may not exchange places in one step, because they would collide in the process, which would definitely lose points for presentation! Also, only one dancer may occupy any one square following any step.
What is the minimum number of steps required to achieve the exchange of men and women?
Question 2558 in JRM 30(4). Answer in JRM 31(4).
In how many ways can we place the digits 1 to 9 into a 3 x 3 array so that each digit is smaller than the digit immediately to its right and is smaller than the digit immediately below it?
Question 2561 in JRM 30(4). Answer in JRM 31(4).
Given a general triangle ABC, construct a chord DE parallel to AB such that DE = AD + EB (see diagram).
Question 2562 in JRM 30(4). Answer in JRM 31(4).
Question 2563 in JRM 30(4). Answer in JRM 31(4).
A square is divided into four triangles, as shown in the diagram. Not counting rotations as different, there are six ways to colour the triangles red, yellow, blue and green, such that each colour is used once in the square.
Is it possible to make a cube from a set of these six squares such that each edge borders two triangles having the same colour — if so, in how many ways can this be done?
Question 2564 in JRM 30(4). Answer in JRM 31(4).
Select any one of the twelve pentominoes (see diagram) and place it on a 5 x 7 board in such a way that the rest of the board can be tiled with ten “right trominoes”.
There are several placements of the P pentomino, one of which is shown here.
You are asked to find (at least) three possible placements of each of the 12 pentominoes. The number of possible placements of trominoes in each case is irrelevant, and rearrangement of the trominoes only does not constitute a different solution.
Question 2565 in JRM 30(4). Answer in JRM 31(4).
A 6 x 6 square has to be cut into eight pieces of any shape with areas 1, 2, 3, 4, 5, 6, 7, and 8 square units.
Question 2566 in JRM 30(4). Answer in JRM 31(4).
Question 2567 in JRM 30(4). Answer in JRM 31(4).
Dissect a unit square into a number of rectangles of different shapes but all having the same area.
Can you find the minimum number of rectangles, and the precise dimensions of each rectangle?
Question 2568 in JRM 30(4). Answer in JRM 31(4). My comment on the solution was accepted for publication in JRM 35(1).
Four mutually touching circles have radii in geometric progression, denoted here by 1, r, r^{2}, r^{3}. What is the value of r?
The geometric progression can be continued indefinitely in each direction — the previous and next circles are shown in the diagram by the black dot and the shaded segment, which have radii 1/r and r^{4}.
Can anyone determine the link between this puzzle and the study of isotopes?
Question 2581 in JRM 31(1). Answer in JRM 32(1).
A large sack contains a number (less than 10000) of balls of nine different colours. If I select two balls at random there is an exactly even chance that they will be of the same colour.
If, instead, I throw away all the balls of one colour, it is still true that if I select two balls at random there is an exactly even chance that they will be of the same colour.
I can continue throwing out all the balls of one colour at a time (until only one colour remains) and it remains true each time that if I select two balls at random there is an exactly even chance that they will be the same colour.How many balls were there originally in the sack
Question 2582 in JRM 31(1). Answer in JRM 32(1).
There are many stadia in Europe with a soccer pitch inside an athletics track (see diagram). can you find the area of the largest possible soccer pitch which can be contained within an athletics track?
For the purposes of this puzzle, assume only that the inside of the track comprises two parallel straights and two semicircles with a total perimeter of 400 meters, and that the soccer pitch is rectangular. In real life there are other constraints, so don’t worry if your solution to the puzzle looks a little odd!
Question 2583 in JRM 31(1). Answer in JRM 32(1).
The ancient Egyptians used fractions in their calculations, but their notation
did not allow them to write fractions such as 11/199, and they did not comprehend
such fractions. They could only use unit numerators, so the result of dividing
11 by 199 might be written, used, and understood as 1/20 + 1/199 + 1/3980.
There is a software package for PCs which transforms proper fractions into
“Egyptian Fractions” — the sum of a series of reciprocals.
For the fraction 50/89, it gives:
50/89 = 1/2 + 1/17 + 1/337 + 1/145681 + 1/29711989585 +
1/1471337208468868797457 +
1/6494499543074890436870241790813851000203090
Can you find something more manageable, with fewer terms and less horrendous denominators?
Question 2584 in JRM 31(1). Answer in JRM 32(1).
In the diagram, the four rows, four columns, and two diagonals, each of which can be read both ways, show 20 different 4digit numbers. Most of them are prime numbers.
Can you construct a fourbyfour grid of digits in which the 20 4digit numbers are all different and all prime?
Question 2585 in JRM 31(1). Answer in JRM 32(1).
An unidentified country has a 7digit population — and everyone has been given a National ID Number, sequentially from one, allocated by no identifiable logic.
The Census Minister has chosen three names at random, and is finding their ID numbers on the computer. When the first number appears on the screen, the Government’s mathematical whizzkid informs the Minister that there is precisely 5050 chance that the other two numbers will both be less than the one just displayed.
What is the population, and what is the first number?
Question 2586 in JRM 31(1). Answer in JRM 32(1).
I have a number of small wooden cubes, all the same size. I arrange them into a cuboid and paint the whole outer surface red. I then arrange the cubes into a different cuboid shape, with no red squares visible, and paint the outer surface white. Finally, I rearrange them again into a cuboid with no painted squares visible, and paint the outer surface blue. I now find that every face of every cube has been painted
If I have the smallest number of cubes, and use all of them each time, how many of them have only two different colours of paint?
Question 2587 in JRM 31(1). Answer in JRM 32(1).
The four triplets of numbers (14, 50, 54), (15, 40, 63), (18, 30, 70), (21, 25, 72) each have the same sum — 118 — and the same product — 37800. Can you find five triplets with this property, with a common product under 100000?
[Misprint in JRM is here corrected.]
Question 2591 in JRM 31(1). Answer in JRM 32(1).
A semiprime is a composite number which has only one pair of divisors. The first few are 4, 6, 9, 10, 14, 15, 21, 22, .... The first trio of consecutive numbers which are all semiprimes is 33—34—35. There are many more such trios, but there cannot be more than three consecutive numbers which are semiprime, and there cannot be more than eight consecutive odd numbers which are all semiprime.
If you can explain why those statements are true, that might help you tackle this problem. Can you find a sequence of four (or more) numbers in arithmetic progression which are all semiprime?
Question 2604 in JRM 31(3). Answer in JRM 32(3).
Cover the maximum possible portion of the area of a square, using four equilateral triangles. The triangles must not overlap each other or extend beyond the sides of the square.
Question 2605 in JRM 31(3). Answer in JRM 32(3).
Consider the equation a^{2} + b^{2} + c^{2} = 10^{n} − 1, where a, b, c, and n are positive integers. We may assume without loss of generality, that a ≤ b ≤ c. The simplest solution is (a, b, c, n) = (1, 2, 2, 1). Find all possible solutions and prove that your list is complete.
Question 2607 in JRM 31(3). Answer in JRM 32(3).
Dissect the five tetrominoes (see below) in the smallest number of pieces which can be reassembled to form a square.
Question 2609 in JRM 31(3). Answer in JRM 32(3).
Determine the values of α for which the function f(x) = 3^{x} + ∣x∣ − α has
Question 2625 in JRM 32(1). Answer in JRM 33(1).
The cross shown in the diagram below differs from those presented by Golomb as numbers 19, 71, 89, and 90 in his book Polyominoes (revised edition, Princeton, 1974).
Construct this cross from the 12 pentominoes.
Assuming that the centre of the X pentomino is in the upper left quadrant, it seemingly may occupy any of the 5 numbered positions. We seek solutions for each of these five possibilities.
Question 2628 in JRM 32(1). Answer in JRM 33(1).
(Note: A misprint in the published question has here been corrected.)
Find all real solutions of the equation
4a^{2} + 6b^{2} + 2c^{2} +
d^{2} = 2 (2ac + ad + bc − bd −
cd)
Question 2629 in JRM 32(1). Answer in JRM 33(1).
An equilateral triangle ABC is folded along the line DE so that its apex A touches the base BC at point F, as shown here.
Question 2650 in JRM 32(3). Answer in JRM 33(3).
The entire surface of an Lshaped smørgåsbord table is covered with dishes of food. The ends of the table are two metres wide, while the outer sides are each 4 metres long: each interior side is thus 2 metres in length. If there is access to a dish from any point along the table perimeter, where is the dish which requires the longest reach and how long is that reach?
Question 2652 in JRM 32(3). Answer in JRM 33(3).
Use each of the digits, 1, 2, 3, 4, 5, 6, 7, 8, and 9 once, in ascending order from left to right, and any of the usual operations — addition, subtraction, multiplication, division and exponentiation — along with parentheses, to represent the number 2004.
[Comment: Almost all the solutions provided, also allowed digits to be simply
catenated, so that the number twelve could be used as simply 12.
I also found a solution to this using the digits in reverse order, from 9 to 1.
Question 2701 in JRM 33(4). Answer in JRM 34(4).
Define a lattice triangle as one whose vertices lie on points in the unit square lattice.
Question 2707, originally posed (but with an error) in JRM 33(4), and corrected in JRM 34(4). It was answered in 35(2)
Triangle ABC is inscribed in a circle. The tangents to the circle at points B and C intersect at D. What is the relation between the sides of the triangle when A and D are at the same distance from line BC? What possible values can be assumed by angle A?
Question 2723 in JRM 34(2). Answer in JRM 35(2).
In the 1690s, it was determined that a heavy, hanging chain assumes the shape
of the catenary curve, which has the equation
y = a cosh(x/a) = (a/2)(e^{(x/a)} +
e^{(−x/a)}).
L. Hodgkin in his A History of Mathematics from Mesopotamia to Modernity (Oxford 2005) reports that Leibniz proposed that ships at sea should have a chain suspended among their instruments, so that, should they lose their tables of logarithms, they could deduce them by measurements on the curve of the chain.
How could they have done this?
[Note that the original question quoted directly from Hodgkin’s work, which I have paraphrased here.]
Question 2724 in JRM 34(2). Answer in JRM 35(2).
For each of the ten decimal digits d, find an expression using only the digits 2, 7, and d exactly once each, and having value 8. Only these three digits and the operators for addition, subtraction, multiplication, division, and exponentiation as well as decimal points and parentheses may be used. For example, 7 – 2 + 3 = 8 is one known solution for d = 3.
Question 2727 in JRM 34(2). Answer in JRM 35(2).
If the two numbers B and C are placed in the positions shown in the 3×3 array in the diagram, there are still an infinite number of ways in which seven additional numbers can be placed so that the sums of the three numbers in each row, column, and diagonal are equal. Nevertheless, given B and C as shown, what must be the value of A?
Question 2728 in JRM 34(2). Answer in JRM 35(2).
In a certain year, the standings in the first round of the FIFA world cup, Group B, were as shown in the table below:
Wins  Losses  Ties  Goals For 
Goals Against 


England  2  0  1  5  2 
Sweden  1  0  2  3  2 
Paraguay  1  2  0  2  2 
Trinidad & Tobago 
0  2  1  0  4 
Knowing that each team plays each of the others exactly once, use the given information to determine the score of each match.
Question 2731 in JRM 34(3). Answer in JRM 35(3).
In the figure A_{0}BC is equilateral and D is midpoint of BC. Let E be any point on the segment BD.
We define a sequence of points A_{n} on A_{0}D by A_{n}B = A_{n−1}E. Find the measure of angle A_{3}ED.
Question 2732 in JRM 34(3). Answer in JRM 35(3).
Arrange the integers 1 through 34 in a sequence such that the sum of every pair of consecutive terms is a Fibonacci number.
Question 2733 in JRM 34(3). Answer in JRM 35(3).
Question 2734 in JRM 34(3). Answer in JRM 35(3).
Let S be a set of points in the Euclidean plane such that each point has at least one neighbouring point at distance 1 unit or less. For each fixed n, determine the maximum possible area of the convex hull of S. The convex hull is the minimum convex set containing all points of S.
Question 2735 in JRM 34(3). Answer in JRM 35(3).
“Amy, how old are you?” asked Bill. Amy replied “Ladies do not like to tell their age, but I will tell you that my age is the smallest integer that can be expressed as the sum of two distinct squares in two different ways.” “How interesting,” said Bill. “My age is the next larger number having the same property.” How old are Amy and Bill?
Question 2737 in JRM 34(3). Answer in JRM 35(3).
A semimagic square is a square matrix in which the sums of all rows and of all columns are equal. Prove or disprove the following: If A and B are two 3 × 3 semimagic squares with magic sums S_{A} and S_{B} respectively, then the matrix product AB is semimagic with magic sum S_{A} × S_{B}.
Question 2738 in JRM 34(3). Answer in JRM 35(3).
Question 2739 in JRM 34(3). Answer in JRM 35(3).
Prove or disprove: If p is a prime greater than 3, then either p^{3} + 1 or p^{3} − 1 is divisible by 18.
Question 2740 in JRM 34(4). Answer in JRM 35(4).
A game is played between Angelica and Basil with a pair of standard cubical unbiased dice. On each throw of the dice, if a double is thrown, Angelica wins. Otherwise, if the total number of pips repeats a total previously thrown in the game, then Basil wins. Otherwise the dice are thrown again. Who has the advantage, and by how much?
Question 2742 in JRM 34(4). Answer in JRM 35(4).
A clock has an hour hand and a minute hand, the latter being longer than the former. At 25 minutes past 2, the distance between the tip of the hour hand and the tip of the minute hand is exactly 161 mm. At 25 minutes to 4, this distance is exactly 199 mm. How far apart are the tips of the hands at 9 o’clock?
Question 2743 in JRM 34(4). Answer in JRM 35(4).
Find an integer, greater than 1, which is the sum of the sixth powers of its digits.
Question 2745 in JRM 34(4). Answer in JRM 35(4).
A rectangular spiral, shaded in the diagram, is a simple closed curve with width of one unit.Find the area and perimeter of the shaded spiral.
Question 2746 in JRM 34(4). Answer in JRM 35(4).
You have two shoelaces and some matches. You are told that each shoelace takes exactly one hour to burn (in fuselike fashion, starting by lighting one end) but that the burning is irregular and possibly different for the two laces, some parts of each lace burning faster than other parts. Explain how to use these laces and matches to time exactly 45 minutes.
Question 2748 in JRM 34(4). Answer in JRM 35(4).
We define an Lshaped polyomino as one consisting of two “arms” projecting at a right angle from a common unit square, each arm being one unit in width. The set of Lshaped polyominoes through size 7 is shown below.
These nine pieces have a total of 50 unit squares. Use the full set to cover the 11 by 5 rectangle with a 5unit central hole, and the 7 by 8 rectangle with a 6unit central hole, which are shown in the diagram below.
Question 2749 in JRM 34(4). Answer in JRM 35(4).
Two points are picked at random in the interval (0,1). What is the probability that one is more then twice the other?
Question 2755 in JRM 35(1). Answer in JRM 36(1).
For each of the following, you may use addition, subtraction, multiplication, division, powers,decimal points, concatenation, and parentheses, but no roots, factorials, or other notations. Find as many solutions as possible for each part.
Question 2758 in JRM 35(1). Answer in JRM 36(1).
A bag initially contains 10 black balls. We repeatedly draw a ball and if it is black, we replace it with a white ball, while if it is white, we return it to the bag. How many repetitions will there be, on average, before all the balls in the bag are white?
Question 2760 in JRM 35(2). Answer in JRM 36(2).
Here are the set of all 24 possible foursided jigsawpuzzle pieces where each side is either straight, has one lobe or one indentation.
Arrange these pieces to form a three by eight rectangle. Pieces may be rotated but not flipped over.
Question 2761 in JRM 35(2). Answer in JRM 36(2).
Generalize the pattern illustrated by the four sums given below.
2^{7} + 1^{7} + 1^{7} =
3 × 2^{6} − 1 × 1 × 2 × 31
3^{7} + 2^{7} + 1^{7} + 1^{7} =
5 × 3^{6} − 1 × 1 × 2 × 31 −
1 × 2 × 3 × 211
5^{7} +3^{7} + 2^{7} + 1^{7} + 1^{7} =
8 × 5^{6} − 1 × 1 × 2 × 31 −
1 × 2 × 3 × 211 −
2 × 3 × 5 × 1441
8^{7} + 5^{7} +3^{7} + 2^{7} + 1^{7} + 1^{7} =
13 × 8^{6} − 1 × 1 × 2 × 31 −
1 × 2 × 3 × 211 −
2 × 3 × 5 × 144 −
3 × 5 × 8 × 9881
In particular, find a general formula for the last number (31, 211, 1441, etc.) in successive equalities.
Question 2770 in JRM 35(3). Answer in JRM 36(3).
Here are the set of all 24 possible foursided jigsawpuzzle pieces where each side is either straight, has one lobe or one indentation.
Arrange one complete set of these pieces to form two separate rectangles. Pieces may be rotated but not flipped over.
Question 2771 in JRM 35(3). Answer in JRM 36(3).
For any given natural number n, ending in the digit 1, 3, or 7, is there a prime number which “ends in n”, i.e. whose rightmost digits are the number n?
Question 2772 in JRM 35(3). Answer in JRM 36(3).
Prove that any palindrome, other than 11, having an even number of digits is not prime.
Question 2773 in JRM 35(3). Answer in JRM 36(3).
For an integer n≥2, we define the Inferior Smarandache Prime part, ISPP(n), as the largest prime less than or equal to n. Similarly, the Superior Smarandache prime Part, SSPP(n), is the smallest prime greater than or equal to n. Solve the Diophantine equation
ISPP(x) + SSPP(x) = k, where k is a positive integer.
Question 2774 in JRM 35(3). Answer in JRM 36(3).
A Generalized Smarandache Palindrome (GSP) is an integer having either of the concatenated forms a_{1}a_{2}...a_{n}a_{n}...a_{2}a_{1} or a_{1}a_{2}...a_{n−1}a_{n}a_{n−1}...a_{2}a_{1}, where the a_{i} are positive integers. We note that if all a_{i} are singledigit integers, then the GSP is palindromic in the usual sense. We also note that every integer is trivially a GSP by letting n = 1 in the second pattern of the definition; we hereby rule out this possibility. Find the number of four digit GSP which are not palindromic.
Question 2781 in JRM 35(4). Answer in JRM 36(4).
For a positive integer n, let f(n) compute the sum of cubes of digits of n, for example, f(325) = 3^{3} + 2^{3} + 5^{3} = 160. Find a solution, other than 1, of f(f(n)) = n.
Question 2783 in JRM 35(4). Answer in JRM 36(4).
It is easy to show that for rational numbers a and b, not both zero, we have (a + b√2)^{−1} = a / (a^{2} − 2b^{2}) + (b√2) / (a^{2} − 2b^{2}). Find an analogous formula for (a + b·2^{⅓} + c·2^{⅔})^{−1}, where a, b, and c are rational numbers, not all zero. Prove that your formula is valid whenever a, b and c are not all zero.
Question 2792 in JRM 36(1). Answer in JRM 37(1).
Let F_{n} denote the n^{th} Fibonacci number. and U(x) denote the units digit of x in base ten. Find k such that the following equality is true for all n.
U(F_{n+6} F_{n+10} F_{n+11} F_{n+13}) = U(F_{n} F^{2}_{n+1} F_{n+k})
Question 2793 in JRM 36(1). Answer in JRM 37(1).
Use one full set of pentominoes (below)
to construct two shapes which can in turn be used to construct either a 6 x 10 rectangle or a 5 x 13 rectangle which encloses a unit square hole and two domino holes (see below).
Question 2794 in JRM 36(1). Answer in JRM 37(1).
Sixteen inhabitants of Mars announce their ages, all of which are different. After a moment of mental calculation one of them notes that if his age is multiplied by that of any of the others, the product is a permutation of the digits in the two ages which were multiplied. We know that Martians use base ten arithmetic and that they can live up to 1000 years. What are the sixteen ages? This problem is offered with an appropriate nod to Problem 2536 (JRM, 30:3, p. 224).
Question 2795 in JRM 36(1). Answer in JRM 37(1).
A pyramid has a rectangular base and four Pythagorean triangle faces. Find such a pyramid in which the four Pythagorean triangles all have different underlying primitive Pythagorean triples.
Question 2796 in JRM 36(1). Answer in JRM 37(1).
Adam Kochansky, a Jesuit mathematician, gave the construction shown below in 1685.
Line RS is tangent at A to the unit circle with centre O. Then with A as centre, draw an arc with unit radius cutting the circle at C. With C as centre, draw another arc with unit radius, cutting the first arc at D. Draw OD intersecting RS at E. Construct EF = 3. Now use the standard square root construction (not shown) to construct a square with side √BF.
Reference: B. Bold, Famous Problems of Geometry and How to Solve Them Dover Publications Inc., New York, 1969.
Question 2797 in JRM 36(1). Answer in JRM 37(1).
Early on a fall morning, I was walking between buildings on the Mount Mercy campus. The sun was directly at my back as I started down a set of stairs. I noticed that as I raised and lowered my feet to move from one step to the next, the shadows cast by my feet on the flat ground at the bottom of the flight of stairs were in the identical position each time. Assuming that all of the steps are identical and that there are n of them, determine the relation between the dimensions of a step and the angle at which the sun’s rays strike the sidewalk.
Question 2799 in JRM 36(1). Answer in JRM 37(1).
The infinite Smarandache Prime Base is {1, 2, 3, 5, 7, 11, 13, 17, ...} (i.e., 1 and all the primes). We express a number in this base making a “greedy” selection of base elements. For example,
16 = 1×13 + 0×11 + 0×7 + 0×5 + 1×3 + 0×2 + 0×1 = 1000100_{SPB}
Question 2824 in JRM 36(4). Answer in JRM 37(4).
Prove: If the points A_{1}, B_{1}, C_{1} divide the sides BC = a, CA = b and AB = c, respectively, of triangle ABC in the ratio k>0, then
AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2} ≥ ¾(a^{2} + b^{2} + c^{2}).
Question 2826 in JRM 36(4). Answer in JRM 37(4).
Each member of a group of people generates a sequence of numbers by the following procedure. First, each person chooses any real number as the initial term of his sequence. Then he calculates the sum of the first terms in all other sequences (not including his own) and uses that sum as his second term. This procedure is repeated for each term, i.e. the (i+1)^{th} term in each sequence is the sum of the i^{th} terms of all other sequences. One person generates a sequence whose first three terms are 1, 3 and 13.
Question 2830 in JRM 37(1). Answer in JRM 38(1).
For a positive integer n, let f(n) be the sum of n and its individual digits, e.g. f(25) = 25 + 2 + 5 = 32. We can generate an increasing sequence by iteration of this function, e.g. 14, 19, 29, 40, 44, ... Then we can partition the set of positive integers according to the number of iterations needed to reach a palindrome. Class 0 consists of the palindromes, Class 1 contains 10, 20, 30, 40, ... etc.
Question 2834 in JRM 37(1). Answer in JRM 38(1).
You may choose to throw any number of standard, unbiased, cubical dice. You win if the total of the numbers rolled on all of the dice is 12. How many dice should you use?
Question 2836 in JRM 37(1). Answer in JRM 38(1).
Consider the Diophantine equation a^{m} + a^{k} = b^{2}. For which values in the range 1 ≤ a ≤ 20 does the equation have a solution? In problem 27 of J. Cofman’s What to solve? (Clarendon, 1990), it is shown that if, for a given value of a there is one solution, then there are, in fact, infinitely many solutions.
Question 2855 in JRM 37(3). Answer in TRM:2.
In the diagram, a circle of radius a is inscribed in a circle of radius 2a so that it is tangent to the larger circle at the point (0, 2a). A circle with centre B is tangent to the larger and smaller circle as well as the xaxis. A circle with centre C is tangent to the circle centred at B and circle centred at (0, a) as well as to the xaxis.
What are the coordinates of the points B and C? This problem was inspired by a wooden decoration at the front of some meeting rooms in the Sheraton Hotel and Towers in Chicago.
Question 2863 in JRM 37(4). Answer in TRM:3.
Find all solutions of the matrix equation .
Question 2866 in JRM 37(4). Answer in TRM:3.
For the quadratic equation f(x) = ax^{2} + bx + c, prove that there is a constant C and an arithmetic sequence (s_{n}) such that for positive integer values n, f(n) is equal to C plus the nth partial sum S_{n} of the arithmetic sequence.
Question 2869 in JRM 37(4). Answer in TRM:3.
In the diagram, if the radius of each inscribed circle is 1, what are the dimensions of the bounding rectangle?
The centres of four of these circles are collinear. (This is problem 12 on p. 107 of F. Swetz's Mathematical Expeditions: Exploring Word Problems Across the Ages, Johns Hopkins, 2012).Question 2871 in JRM 38(1). Answer in TRM:4.
The figure below, a solution to problem 2822 (JRM 36(4), p. 359), shows that the Zpentomino and three others can be used to construct a double scale version of itself.
We readily see that the Xpentomino does not share this property. What about the other ten pentominoes?
Question 2874 in JRM 38(1). Answer in TRM:4.
We can express the number 1 in several ways as the sum of two fractions which together use each of the nine nonzero digits exactly once. The fractions are not necessarily in lowest terms. For example, 1 = ^{4}⁄_{12} + ^{638}⁄_{957}.
Question 2875 in JRM 38(1). Answer in TRM:4.
Show that every positive integral power of √2  1 can be expressed in the form √m  √(m1).
Question 2876 in JRM 38(1). Answer in TRM:4.
H. E. Dudeney in his Amusements in Mathematics, question 90, “The Century Puzzle”, asks us to represent the number 100 as a “mixed fraction”, that is of the form A + ^{B}⁄_{C}, using each of the digits 1 – 9, once only. For example 100 = 82 + ^{3546}⁄_{197}. He says that Edouard Lucas had found seven solutions, but in fact there are 11, and in that Dudeney was correct. There are 11 solutions, which he supplies in his answers.
Question 2877 in JRM 38(1). Answer in TRM:4.
In his book, Mathematics Galore (MAA, Washington DC), James Tranton poses the problem of finding a number N such that all of the multiples N, 2N, ..., 10N contain the digit 3.
Question 2879 in JRM 38(1). Answer in TRM:4.
We consider the inequality aπ + be + cφ < 10^{6} where π, e, and φ (golden ratio) are mathematical constants, while a, b, and c are integers. Find solutions where abc ≠ 0, and
Question 2880 in JRM 38(2). Answer in TRM:8.
The ordered set of odd numbers beginning with the integer 3 is a list that contains all the prime numbers greater than 2. It has half as many integers as does the ordered set of all integers beginning at 3. One process for generating the elements of the list of odd numbers is to begin with the number 3 and repeatedly add 2.
Define a similar process that generates a list of integers that contains all the prime numbers beginning with prime P and has fewer than onefourth as many integers as the ordered set of all integers beginning at P. As in the example given above, your list may contain integers which are not prime.
Question 2883 in JRM 38(2). Answer in TRM:8.
A circular clock face has radius 12. The hour hand and minute hand have zero thickness and move continuously at a constant speed, not in discrete jumps. At 12:00 the hands coincide, of course. At how many minutes after 12:00 will they be positioned so that a disk of radius 2 can be tangent to each of them and to the circumference of the clock?
Question 2884 in JRM 38(2). Answer in TRM:8.
Question 2885 in JRM 38(2). Answer in TRM:8.
For each of the following, you may use addition, subtraction, multiplication, division, powers, decimal points, concatenation, and parentheses, but no roots, factorials, or other notations. Find as many solutions as possible for each part.
Question 2886 in JRM 38(2). Answer in TRM:8.
Question 2887 in JRM 38(2). Answer in TRM:8.
Question 2889 in JRM 38(2). Answer in TRM:8.
The pancake flipping problem was mentioned in a presentation by Ivars Peterson at the Iowa section meeting of the Mathematical Association of America. It has a rich history and has applications in computer network theory.
A cook is incapable of cooking a batch of pancakes where all of them are the same size. Once a stack is complete, they will be stacked in a random order and the waiter will reorganize the stack so that they go from the largest to the smallest from the bottom to the top. This is accomplished by performing a flip, which is placing a spatula somewhere in the pile and flipping the entire stack of pancakes above the spatula over.
Problem: Assuming that it takes at most k flips to orient a stack of n pancakes, prove that it will take at most k+2 flips to orient a stack of n+1 pancakes.
Question 1 in TRM:1. Answer in TRM:5.
Prove that, for a suitably chosen integer x_{1}, the recursive formula x_{n+1} = x_{n} + 4 + 4√(x_{n}  a), where a is a positive integer, produces an infinite sequence of integers.
Question 2 in TRM:1. Answer in TRM:5.
On page 188 of The Gödelian Puzzle Book: Puzzles, Paradoxes & Proofs,
Raymond M. Smullyan describes dyadic notation, where all integers greater than
zero can be expressed using a string of 1‘s and 2‘s. Using the
alphabet {1,2} all such integers can be expressed in the form
2^{n}d_{n} +
2^{n1}d_{n1} + ... +
2^{1}d_{1} + 2^{0}d_{0}
For example, 4, 5 and 6 are 12, 21, and 22 in dyadic form.
Shortly after the description of dyadic notation there is the sentence: “It is just that for any given number in decimal notation, the digits of the dyadic and binary expressions of the number are different.”
Question 3 in TRM:1. Answer in TRM:5.
In the figure, a circle of radius a is inscribed in a circle with radius 2a and centre (0,0), so that it is tangent to the larger circle at the point (2a, 0). A circle with centre B(b, d) is tangent to the other two circles as well as to the xaxis. Finally, a circle with centre C(c, e) is tangent to all three circles mentioned previously.
Find the centre and radius of this last circle.
Question 4 in TRM:1. Answer in TRM:5.
In the figure, a circle with centre A(a, 0) is inscribed in a circle radius 2a and centre (0, 0), so that it is tangent to the larger circle at point (2a, 0). A circle with centre B(b,d) is tangent to these other two circles as well as to the xaxis. Finally, a circle with centre C is tangent to the xaxis and to the two circles with centres A and B.
Find the centre and radius of this last circle.
Question 9 in TRM:1. Answer in TRM:5.
Four students (A, B, C and D) enter a threeday mathematics competition and each is assigned two integers to be used throughout. A is assigned 2 and 9, B gets 3 and 8, C gets 4 and 7, while D has 5 and 6. Each day they are paired in teams of two and the pairings are never repeated. Each day the team’s task is to use their four digits to form expressions which equal 22. They may concatenate digits as well as using the operators +, –, ×, ÷ and √. Each expression must use all four of their digits and operators may be used more than once in any expression. The radical symbol may not have an index, so it indicates a square root only.
Question 12 in TRM:1. Answer in TRM:5.
Find a fourpiece dissection of a 7 × 7 square and a 24 × 24 square such that the pieces can be reassembled to form a 25 × 25 square. Pieces may be turned over but cuts may only be made along boundaries of the unit squares which make up the larger squares.
Question 4 in TRM:3. Answer in TRM:8.
Let ABCD be a square. Let E be the midpoint of BC, F the midpoint of CD and G the midpoint of BE. Prove that the lines AE, BF and DG are concurrent.