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Nov 9 19

by Brian Gladman

Sunday Times Teaser 2981 – Faulty Pedometer

by Howard Williams

Published November 10 2019 (link)

Judith is a keen walker who uses a five-digit pedometer to record her number of steps. Her pedometer is inaccurate as some of the counters consistently move on to 0 early by missing out one or more digits. For instance, one of them might roll over from 7 to 0 every time instead of from 7 to 8, missing out digits 8 and 9. She is, however, well aware of this and can work out the correct number of steps.

After walking her usual distance, the pedometer shows 37225 steps but she knows that the true number is 32% less than this. A second distance she walks requires a 30% reduction in the number displayed to give the true number of steps.

How many steps is the second distance?

Nov 2 19

by Brian Gladman

Sunday Times Teaser 2980 – Egyptian Weights and Measures

by Susan Bricket and John Owen

Published November 3 2019 (link)

We were wondering why ancient Egyptians chose to represent arbitrary fractions as sums of distinct unit fractions of the form 1/n (thus 5/7 = 1/2+1/5+1/70). One of us recalled long ago watching our greengrocer use four brass weights of 1/2, 1/4, 1/8, 1/16 lb to weigh any number of ounces up to 15 by stacking some of them on one side of her balancing scales. We wanted to make a metric equivalent, a set of distinct weights of unit fractions of a kilo, each weighing a whole number of grams, to weigh in 10g steps up to 990g.

Naturally, we wanted to use as little brass as possible, but we found that there were several possible such sets. Of these, we chose the set containing the fewest weights.

List, in increasing order, the weights in our set.

Oct 26 19

by Brian Gladman

New Scientist Enigma 987 – Two Halves Make a Whole

by Richard England

From New Scientist #2142, 11th July 1998

A semi-prime is the product of two prime numbers; the square of a prime counts as a semi-prime.

Harry, Tom and I were trying to find pairs of 2-digit semi-primes such that if we added the two semi-primes together we formed a 2-digit prime. We each found three such pairs; the 18 semi-primes we used and the 9 primes that were formed were all different.

Harry’s three odd semi-primes were all greater than 50; Tom’s three even semi-primes were all greater than 50.

What were my three pairs of semi-primes?

Oct 26 19

by Brian Gladman

Sunday Times Teaser 2979 – Mischievous Sam

by Graham Smithers

Published October 27 2019 (link)

I set Sam a question, the answer to which was a 3-digit number, with the digits increasing by 1 from first to last (eg 789)

Sam eventually produced a 3-digit answer, but only 2 of his digits were correct and in the correct position. The third digit was wrong.

Investigating further I found that Sam had the correct answer but, for devilment, decided to change it into a different (single-digit) base.

If I were to tell you which of his 3 digits was the wrong one, you should be able to tell me:

(a) the correct answer, and
(b) the base used by Sam.

Oct 19 19

by Brian Gladman

Sunday Times Teaser 2978 – Norfolk Flats

by Andrew Skidmore

Published October 20 2019 (link)

Sam has purchased Norfolk Flats; an area of farmland (less than 100 hectares) bordered by six straight fences. He intends to farm an area which is an equilateral triangle with corners that are the midpoints of three of the existing boundaries. This creates three more distinct areas (one for each of his sons); these areas are identical in shape and size and have two sides that are parallel.

Sam measured the area (in square metres) which each son will farm and also his own area. One of the numbers is a square and the other a cube. If I told you which was which, you should be able to work out the area of Norfolk Flats.

What (in sq metres) is that area?

Oct 13 19

by Brian Gladman

New Scientist Back Page Puzzle 24 – Three Stamps

by David Bodycombe

From New Scientist #3250, 5th October 2019

I’m on holiday in the lovely country of Philitaly, and planning to send plenty of postcards because postage is very cheap. But the country only allows up to three stamps on any letter.

Can you tell me which three denominations of stamps would allow me me to cover any cost of postage from 1 cent to 15 cents inclusive?

And which four stamp denominations would allow all values from 1 to 24 cents?

Oct 13 19

by Brian Gladman

New Scientist Back Page Puzzle 23 – Circling the Squares

by Rob Eastaway

From New Scientist #3249, 28th September 2019

Darts player Juan Andred has noticed that on a standard dartboard, there are some neighbouring pairs of numbers that add up to a square number. For example, 20 and 5 make 25, while 6 and 10 add up to 16. He has been wondering if he can come up with a new arrangement of the numbers 1 to 20 so that all neighbouring pairs add up to a square number. And he has nearly succeeded.

He has 20 at the top of the board, and every pair of neighbours adds to a square — with one exception. On his new board, 18 doesn’t form a square with its clockwise neighbour, which is 15, or with its anticlockwise neighbour.

What does Juan’s “square” dartboard look like?

Oct 12 19

by Brian Gladman

New Scientist Enigma 989 – Prime Progressions

by Richard England

From New Scientist #2144, 25th July 1998

Harry and Tom have been investigating sets of positive integers that form arithmetic progressions where all the members of the set are prime numbers.

When they looked for a set of five such prime numbers, Harry found the set whose final (largest) prime is the smallest possible for the final prime in such a set. Tom found the set whose final (largest) prime is the next smallest possible.

Exactly the same thing happened when they looked for a set of six such prime numbers, and again when they looked for a set of seven such prime numbers.

What were the smallest and the largest primes in each of the three sets that Tom was able to find?

Oct 12 19

by Brian Gladman

Sunday Times Teaser 2977 – Enjoying Retirement

by Danny Roth

Published October 13 2019 (link)

George and Martha have worked on separate departments of a company which has four-digit telephone extensions. George looked at his extension and it was abcd. Martha’s (larger) also had a, b and c as her first three digits but not necessarily in that order. Her last digit was e. They added up their two four-digit numbers and found that the least significant digit was f. They then looked at the difference and that was a four-digit number of which the least significant digit was g. They then looked at the product and the least significant digit was h. They then looked at the average of the extensions; in it the first two digits were equal, the last two digits were also equal, and the least significant digit was i. I have thus mentioned nine digits, all positive and unequal.

What was Martha’s extension?

Oct 5 19

by Brian Gladman

Sunday Times Teaser 2976 – Piece of Cake

by Stephen Hogg

Published October 6 2019 (link)

Using her special recipe, so that each cubic inch of baked cake weighed one ounce, Mam made the cake for my eighth birthday party. It had a regular octagonal flat top and base, and equal square vertical faces, several inches high exactly.

Not all (but a majority) of the dozen invited pals came to the party. We each had an equal portion of cake (the largest whole number of ounces possible from it, a two-figure number). Mam had the leftover cake. Curiously, if one more or one fewer pal had turned up and our portions had been worked out in the same way, Mam’s leftover would have been the same in each case, but less than she actually got.

How large, in ounces, was my portion?