Hannah, Joan and Sarah each live at a different one of the three houses numbered l, 2, 3. Alan, Michael and Peter each painted a different one of those three houses. Each of the three women met a different one of the three men at the shops.

There are four clues as to who lives where and so on. However, as the clues are very long they are given here only in a condensed form, and so you may wish first to write them out in full, as indicated:

(1) Write out “the man who painted the house of the woman who met, at the shops,” (1988 + the number of the house of the woman met by Peter) times. Write the copies of the repeated phrase one after the other, and then, in from of the first copy, write “Alan is” and, after the last copy, write “Alan.” to make one very long sentence which is the first clue.

(2) This is similar to clue (1), but has “Michael is” at the start and “Peter.” at the end.

(3) Write out “the woman who met, at the shops, the man who painted the house of” (1066 + the number of the house painted by the man who met Hannah) times. Add “Hannah is” at the start and “Joan.” at the end.

(4) This is similar to clue (3) but the repeated phrase is “the house that was painted by the man who met, at the shops, the woman who lives at”. The clue starts with “Number 1 is” and ends with “Number 2.”

What are the facts, that is, who lives where, who painted which house, who met who?

]]>A right regular prism has two ends with identical faces, joined by oblong rectangular faces. I have eight of them, with regular convex polygonal end-faces of 3, 4, 5, 6, 7, 8, 9 and 10 sides (triangle, square and so on). They sit on my flat desk (on oblong faces), and each prism has the same height.

I chose three prisms at random, and was able to slide them into contact, broadside, in such a way that the middle one overhung both others (and could be lifted without disturbing them). Also, I was able to slide one outer prism to the other side, and the new “middle” prism was overhung by both others (and so vertically “imprisoned” by them).

I was able to do all this again with three randomly chosen remaining prisms.

Give the prior chance of this double selection (as a fraction in lowest terms)

]]>Without repeating a digit I have written down three numbers, all greater than one.

Each number contains a different number of digits.

If I also write down the product of all three numbers, then the total number of digits I have used is ten.

The product contains two different digits, each twice, neither of which appear in the three original numbers.

What is the product?

]]>I wrote down a 2-digit number and a 5-digit number and then carried out a long division.

I then erased all of the digits in the calculation, other than one of them, which I have indicated by #.

This gave me the image above.

What were my two original numbers?

]]>Yesterday was my grandson’s birthday and we continued a family tradition. I asked him to use any eight different non-zero digits (once each) to form a set of numbers that added to 2019. Last year I asked the equivalent question with a sum of 2018, and I have done this each year for over ten years. Only on one occasion has he been unable to complete the task.

In this year’s answer his set of numbers included a 3-figure prime that had also featured in last year’s numbers.

(a) In which year was he unable to complete the task?

(b) What was the 3-figure prime that featured in this year’s answer?

]]>To mark her 55th birthday, Martha, a school teacher, gave each of her nine pupils a sheet of paper with a single different digit from 1 to 9 written on it.

They stood at the front of the classroom in a row and the 9-digit number on display was divisible by 55. Martha then asked the first 3 in the row (from the left) to sit down. The remaining 6-digit number was also divisible by 55. The next 3 then sat down and the remaining 3-digit number was also divisible by 55.

The 9 digit number was the smallest possible. What was it?

]]>Adam and Eve have convex pentagonal gardens consisting of a square lawn and paving. Both gardens have more than one right-angled corner. All the sides of the gardens and lawns are the same whole number length in metres, but Adam’s garden has a larger total area. Eve has worked out that the difference in the areas of the gardens multiplied by the sum of the paved areas (both in square metres) is a five-digit number with five different digits.

What is that number?

]]>The above is the size of the evil multitude in the last “Infernal indices” novel “A deficit of daemons”.

“Spoiler alert” — at the end, the forces of good engage in the fewest simultaneous battles that prevent this evil horde splitting, wholly, into equal-sized armies for that number of battles. The entire evil horde is split into one army per battle, all equally populated, bar one which has a deficit of daemons, leading to discord and a telling advantage for the forces of good.

How many battles were there?

]]>Oliver lives with his parents, Mike and Nellie, at number 5. In each of numbers 1 to 4 lives a family, like his, with a mother, a father and one child. He tries listing the families in alphabetical order and produces the table below.

House Number |
1_______ |
2_______ |
3_______ |
4_______ |
5_______ |

Fathers |
Alan | Dave | George | John | Mike |

Mothers |
Beth | Ellen | Helen | Kate | Nellie |

Children |
Carole | Freddie | Ingrid | Larry | Oliver |

_______

However, apart from his own family, there is just one father, one mother and one child in the correct position. Neither Helen nor Beth lives at number 3 and neither Dave nor Ingrid lives at number 1. George’s house number is one less than Larry’s and Beth’s house number is one less than Carol’s.

Apart from Oliver’s family, who are correctly positioned?

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