Teaser 3221

by Victor Bryant

Published Friday June 14 2024

Faulty Towers

The famous “Tower of Hanoi” puzzle consists of a number of different-sized rings in a stack on a pole with no ring above a smaller one. There are two other poles on which they can be stacked and the puzzle is to move one ring at a time to another pole (with no ring ever above a smaller one) and to end up with the entire stack moved to another pole.

Unfortunately I have a faulty version of this puzzle in which two of the rings are of equal size. The minimum number of moves required to complete my puzzle (the two equal rings may end up in either order) consists of four different digits in increasing order.

What is the minimum number of moves required?

Teaser 3220

by Angela Newing

Published Friday June 07 2024

Graffiti

Six pupils were in detention having been caught painting graffiti on the school wall. The teacher decided to show them some examples of abstract art and produced seven different pictures in order ABCDEFG for them to study for a few minutes. Teacher took them back, showed them again upside down and in random order, and asked them to write down which of ABCDEFG each one was.

The answers were:

Helen	A B C D E F G
Ian	A B C F E D G
Jean	A D F G C E B
Kevin	A D B F C E G
Lee	A F C G E D B
Mike	C A F B E D G

It transpired that each picture had been correctly identified by at least one pupil, and everyone had a different number of correct answers.

What was the correct order for the upside down ones?

Teaser 3219

by Howard Williams

Published Friday May 31 2024

Number Funnel

Betty, being a keen mathematician, has devised a game to improve her children’s maths additions. She constructed an inverted triangular tiling of hexagonal shapes, with nine hexagons in the top row, eight in the second row etc, reducing to one at the bottom. Then, completing the top row with 0s and 1s, she asks them to complete each row below by adding in each hexagon shape the numbers in the two hexagons immediately above it.

In the last game she noticed that if the top row is considered as a base-2 binary number, then this is exactly four and a half times the bottom total.

What is the bottom total?

Teaser 3218

by Stephen Hogg

Published Friday May 24 2024

Algorithm Al

On his 35th birthday, maths teacher Al’s three younger half-sisters bought him “The Book of Numbers for Nerds” as a tease. It showed how to find right-angle triangles with whole-number sides using any two unequal odd square numbers. You take half their sum; half their difference; and the square root of their product to get the three sides. Any multiple of such a triplet would also work. He told his sisters this and that their ages were the sides of such a triangle. “Algorithm Al!” they yelled.

Knowing the age of any one sister would not allow you to work out the other ages with certainty, but in one case you could be sure of her place chronologically (youngest, middle or oldest).

Give the three sisters’ ages (youngest to oldest).

Teaser 3217

by Danny Roth

Published Friday May 17 2024

Peace in Rest

George and Martha regularly read the obituaries in the national press; as well as the date of death, the date of birth of the person discussed is always shown. “That is interesting!” commented Martha as she looked at the three obituaries displayed one morning. Two of them have palindromic dates of birth (eg, 13/11/31, 21/6/12). “Very unlikely, indeed!” agreed George.

Assuming that birth dates are expressed as day (1-31), month (1-12) year (00-99), what is the probability of a palindromic birth date in the 20th century (1900 to 1999 inclusive), and will it be greater, equal or less in the 21st century?

Teaser 3216

by Susan Bricket

Published Friday May 10 2024

Quel Carve-up!

A French farmer’s estate is shaped like a right-angled triangle ABC on top of a square BCDE. The triangle’s hypotenuse is AB, and its shortest side, AC, has length 1 kilometre. Nearing retirement, the farmer decides to sell off the square of land and, obeying the Napoleonic law of succession, divide the triangle into three equal plots, one for each of his two children and a third for him and his wife in retirement. His surveyor discovers, surprisingly, that his remaining triangle of land can be divided neatly into three right-angled triangles, all identical in shape and size (allowing for reflections / rotations).

How many hectares did the farmer sell? (1 hectare = area of 100m x 100m plot)

Teaser 3215

by Andrew Skidmore

Published Friday May 03 2024

Darts League

In our darts league, each team plays each other once. The result of each match is decided by the number of legs won, and each match involves the same number of legs. If both teams win the same number of legs, the match is drawn. The final league table shows the games won, drawn or lost, and the number of legs won for and against the team. The Dog suffered the most humiliating defeat, winning only one leg of that match. Curiously, no two matches had the same score.

What was the score in the match between The Crown and The Eagle?

Teaser 3214

by Peter Good

Published Friday April 26 202

Squaring the Square

Clark took a sheet of A4 paper (8.27 × 11.69 inches) and cut out a large square with dimensions a whole number of inches. He cut this into an odd number of smaller squares, each with dimensions a whole number of inches. These were of several different sizes and there was a different number of squares of each size; in fact, the number of different sizes was the largest possible, given the above.

It turns out that there is more than one way that the above dissection can be made, but Clark chose the method that gave the smallest number of smaller squares.

How many smaller squares were there?

Teaser 3213

by Colin Vout

Published Friday April 19 2024

A Streetcar Named Divisor

In retrospect it was inadvisable to ask an overenthusiastic mathematician to overhaul our local tram routes. They allocated a positive whole number less than 50 to each district’s tram stop. To find the number of the tram going from one district to another you would “simply” (their word, not mine) find the largest prime divisor of the difference between the two districts’ numbers; if this was at least 5 it was the unique route number, and if not there was no direct route.

The routes, each in increasing order of the stop numbers, were: Atworth, Bratton, Codford; Atworth, Downlands, Enford; Bratton, Figheldean, Enford; Downlands, Figheldean, Codford; Codford, Enford.

What were the route numbers, in the order quoted above?

Teaser 3212

by Victor Bryant

published Friday April 12 2024

Changes at Chequers

I started with a six-by-six grid with a 0 or 1 in each of its 36 squares: they were placed in chequerboard style with odd rows 101010 and even rows 010101. Then I swapped over two of the digits that were vertically adjacent. Then in three places I swapped a pair of horizontally adjacent digits.

In the resulting grid I read each of the six rows as a binary number (sometimes with leading zeros) and I found that three of them were primes and the other three were the product of two different primes. The six numbers were all different and were in decreasing order from the top row to the bottom.

What (in decimal form) were the six decreasing numbers?