If you rearrange the letters of a word to form another word, it’s called an anagram. So I think that if you rearrange the digits of a number to form another number, it could be called an anarithm.
Some decimal numbers when expressed in two number bases have representations that are anarithms of each other. For instance, the decimal number 66 is 123 in base 7 and 231 in base 5.
I’ve recently been looking at numbers in base 8 and base 5 and found that it is possible to have a decimal number whose representations in base 8 and base 5 are anarithms of each other.
In decimal notation, what is the largest such number?
Delia plays a game with a stack of different pieces of card. To shuffle the stack she deals them quickly into four equal piles (face-downwards throughout) — the top card starts the first pile, the next card starts the second pile, then the third starts the third pile and likewise the fourth. She continues to place the cards on the four piles in order. Then she puts the four piles back into one big stack with the first pile on the bottom, the second pile next, then the third, with the fourth pile on the top.
She is very pleased with her shuffling method because each card moves to a different position in the stack, so she decides to repeat the shuffle a few more times. However, this is counter-productive because after fewer than six such shuffles the cards will be back in their original order!
How many cards are there in her stack?
George and Martha’s five daughters left home some years ago but have all moved into the same road, Square Avenue, which has 25 normally numbered houses. At Christmas last year, the parents drove to the road to visit but, with age rolling on, George’s memory was beginning to fail him. “Can’t remember the house numbers!” he moaned. “Quite easy!” retorted Martha, “if you remember three things. If you take the house numbers of Andrea, Bertha and Caroline, square each of them and add up the three squares, you get the square of Dorothy’s house number. Furthermore, the average of Andrea’s and Dorothy’s house numbers equals the average of Bertha’s and Caroline’s house numbers. Finally, Elizabeth’s house number is the average of the other four and could not be smaller.”
Which five houses got a knock on the door?
Andrew was building a rhombus-shaped patio. He could have paved it with equilateral triangular slabs with 1ft edges, but he wanted a layout with a hexagonal slab in place of six triangular ones at various places. He considered two layouts that were symmetrical in two perpendicular directions:
Layout 1: every triangular slab touches the perimeter of the patio in at least one point.
Layout 2: each group of three adjacent hexagonal slabs encloses exactly one triangular one.
The triangular and hexagonal slabs come in boxes of 12, and Andrew chose layout 1 because he would only need a third as many boxes of triangular slabs as layout 2.
What length were the sides of the patio and how many slabs in total would be left over?
Four ramblers came to a tunnel that they all had to travel through. The tunnel was too dangerous to travel through without a torch, and unfortunately they only had one torch. It was also too narrow to walk through more than two at a time. The maximum walking speed of each of the walkers was such that they could walk through the tunnel in an exact number of minutes, less than ten, which was different for each walker. When two walkers walked together, they would walk at the speed of the slower one.
They all managed to get through the tunnel and in the quickest possible time, this time being five sixths of the total of their individual crossing times.
In ascending order, what are their four individual crossing times?
Phil buys a four-figure combination lock where each dial can be set from 0 to 9, and needs to decide what number to set to unlock it. He opts for a number with all different digits and a leading zero so he only has to remember a three-figure number. When the chosen number is set for the lock to open, nine other four-figure numbers are visible. For instance, if he chooses 0123, then 1234, 2345, 3456, 4567, 5678, 6789, 7890, 8901 and 9012 are all visible. As a retired Maths teacher with a natural interest in numbers he examines the other nine numbers that are displayed when the lock is set to open. None of these numbers are prime but two of them are perfect squares.
What three-figure number does Phil need to remember?
The PE game W-hoops! involves throwing inflexible hoops to hook onto a W-shaped frame. The torus-shaped hoops all have the same cross-section but different diameters and fit snugly, in contact, into a storage box. The illustration (not to scale) shows side- and top-views with three hoops, but the box actually contains the maximum possible number of hoops, allowing for a hole at the centre. The internal width of the box is a multiple of its internal depth.
Di preferred maths to PE and after her throws she calculated, using 22/7 for pi, that the total volume of the hoops was over 60 per cent of the internal volume of their box. Knowing this would overstate the value, she then used 3.14 for pi and calculated a value under 60 per cent.
How many hoops does the box hold?
My encyclopaedia consists of many volumes, each containing the same number (over 40) of leaves. (Each leaf has a page number on each side. With X leaves Volume 1 would have pages 1 to 2X, Volume 2 would have pages 2X+1 to 4X, etc.) I keep the encyclopaedia on one long shelf so that on their spines I can read “Volume 1”, “Volume 2”, etc, from left to right.
A voracious bookworm started at the left-hand end of the shelf and ate through some covers and leaves, finishing by eating through the leaf with a page number double the number of leaves it had eaten through. Meanwhile, another bookworm started at the right-hand end of the shelf and ate through twice as many leaves as the first bookworm. Then in two of the volumes the percentage of nibbled leaves was equal to the volume number.
How many volumes are there in the set, and how many leaves per volume?
WUQRVWQRXWTRUQWRVQWTSRQWVRSTWQVRWQURTWXRQVWRTSX
Skaredahora’s venture into atonal music — music with no fixed key — had as its (so-called!) melody a long sequence employing 8 different notes, which he labelled Q to X. Behind his composition were 7 keys each using 4 notes. Every three consecutive melody notes obeyed specific rules: all three had to be different; they had to belong to exactly one of the keys; they had not to repeat in the same order any other three consecutive notes; and the key had to change at every step. His chosen melody was as above.
What were the keys involving X, in alphabetical order individually and collectively (eg, QRTX, SUVX)?
Millions of people will today turn their thoughts to those throughout the world whose lives are made miserable by the ravages of war and bitter conflict. I have taken a selection of letters from the alphabet and given each a different single-digit value. In this way the two words that form the title of this teaser represent two six-figure numbers, one of which is a factor of the other.
There are two letters in EASTER PRAYER for which the following is true. If I told you the value of that letter alone, you wouldn’t be able to work out all the others, but if I told you both their values then you could work out all the values.
What is the value of PRAYER?