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Oct 10 20

by Brian Gladman

Sunday Times Teaser 3029 – Square Jigsaw

by Victor Bryant

Published Sunday October 11 2020 (link)

I chose a whole number and asked my grandson to cut out all possible rectangles with sides a whole number of centimetres whose area, in square centimetres, did not exceed my number. (So, for example, had my number been 6 he would have cut out rectangles of sizes 1×1, 1×2, 1×3, 1×4, 1×5, 1×6, 2×2 and 2×3.) The total area of all the pieces was a three-figure number of square centimetres.

He then used all the pieces to make, in jigsaw fashion, a set of squares. There were more than two squares and at least two pieces in each square.

What number did I originally choose?

Oct 3 20

by Brian Gladman

Sunday Times Teaser 3028 – Rainbow Numeration

by Stephen Hogg

Published Sunday October 04 2020 (link)

Dai had seven standard dice, one in each colour of the rainbow (ROYGBIV). Throwing them simultaneously, flukily, each possible score (1 to 6) showed uppermost. Lining up the dice three ways, Dai made three different seven-digit numbers: the smallest possible, the largest possible, and the “rainbow” (ROYGBIV) value. He noticed that, comparing any two numbers, only the central digit was the same and each number had just one prime factor under 10 (different for each number).

Hiding the dice from his sister Di’s view, he told her what he’d done and noticed, but wanted her to guess the “rainbow” number digits in ROYGBIV order. Luckily guessing the red and orange dice scores correctly, she then calculated the others unambiguously.

What score was on the indigo die?

Sep 25 20

by Brian Gladman

Sunday Times Teaser 3027 – Long Shot

by Andrew Skidmore

Published Sunday September 27 2020 (link)

Callum and Liam play a simple dice game together using standard dice (numbered 1 to 6). A first round merely determines how many dice (up to a maximum of three) each player can use in the second round. The winner is the player with the highest total on their dice in the second round.

In a recent game Callum was able to throw more dice than Liam in the second round but his total still gave Liam a chance to win. If Liam had been able to throw a different number of dice (no more than three), his chance of winning would be a whole number of times greater.

What was Callum’s score in the final round?

Sep 18 20

by Brian Gladman

Sunday Times Teaser 3026 – Party Time

by Graham Smithers

Published Sunday September 20 2020 (link)

A four-digit number with different positive digits and with the number represented by its last two digits a multiple of the number represented by its first two digits, is called a PAR.

A pair of PARs is a PARTY if no digit is repeated and each PAR is a multiple of the missing positive digit.

I wrote down a PAR and challenged Sam to use it to make a PARTY. He was successful.

I then challenged Beth to use my PAR and the digits in Sam’s PAR to make a different PARTY. She too was successful.

What was my PAR?

Sep 13 20

by Brian Gladman

Sunday Times Teaser 3025 – Please Mind the Gap

by Howard Williams

Published Sunday September 13 2020 (link)

Ann, Beth and Chad start running clockwise around a 400m running track. They run at a constant speed, starting at the same time and from the same point; ignore any extra distance run during overtaking.

Ann is the slowest, running at a whole number speed below 10 m/s, with Beth running exactly 42% faster than Ann, and Chad running the fastest at an exact percentage faster than Ann (but less than twice her speed).

After 4625 seconds, one runner is 85m clockwise around the track from another runner, who is in turn 85m clockwise around the track from the third runner.

They decide to continue running until gaps of 90m separate them, irrespective of which order they are then in.

For how long in total do they run (in seconds)?

Sep 12 20

by Brian Gladman

New Scientist Enigma 570 – Dead Right

by Susan Denham

From Issue #1723, 30th June 1990

Two friends play a number game related to a popular board-game. Alan  thinks of a secret four-figure number (that is, between 1000 and 9999) and Brian has to guess what it is. Each of Brian’s guesses is ‘marked’ by Alan who tells Brian how many of the digits are ‘dead-right’ (that is, correct and in the correct place) and how many others are ‘misplaced’ (that is, correct but in the wrong place). For example if the secret number were 4088 then the guess 4840 would get the marks ‘one dead-right, two misplaced’. Brian has to find the secret number with as few guesses as he can.

In a recent game Brian’s first few guesses and their ‘marks’ were:

First guess: 1234. One dead-right, one misplaced
Second guess: 1355. One dead-right, none misplaced.
Third guess: 1627. None dead-right, one misplaced.

For his fourth guess Brian chose the lowest number which could still be the secret number but it had none dead-right. I forget how many it  had ‘misplaced’ but from the marks for the fourth guess Brian was able to work out what the secret number was.

What was it?

Sep 4 20

by Brian Gladman

Sunday Times Teaser 3024 – Triple Jump

by Bill Kinally

Published Sunday September 06 2020 (link)

From a set of playing cards, Tessa took 24 cards consisting of three each of the aces, twos, threes, and so on up to the eights. She placed the cards face up in single row and decided to arrange them such that the three twos were each separated by two cards, the threes were separated by three cards and so forth up to and including the eights, duly separated by eight cards. The three aces were numbered with a one and were each separated by nine cards. Counting from the left, the seventh card in the row was a seven.

In left to right order, what were the numbers on the first six cards?

Aug 28 20

by Brian Gladman

Sunday Times Teaser 3023 – Timely Coincidence

by Danny Roth

Published Sunday 30th August 2020 (link)

George and Martha possess two digital “clocks”, each having six digits. One displays the time on a 24-hour basis in the format hh mm ss, typically 15 18 45, and the other displays the date in the format dd mm yy, typically 18 07 14.

On one occasion, George walked into the room to find that the two “clocks” displayed identical readings. Martha commented that the long-term (400-year) average chance of that happening was 1 in just over a six-digit number. That six-digit number gives the birth date of one their daughters.

On what date was that daughter born?

Aug 22 20

by Brian Gladman

New Scientist Puzzle 73 – Changing Guard

by Rob Eastaway

From Issue #3296, 22nd August 2020 (link)

Fifteen members of the King’s 99th Dragoons are standing on parade. “Right turn!” screams the sergeant, and each soldier makes a 90° turn. Unfortunately, many of the squad struggle to know their left from their right. After this manoeuvre, five soldiers end up facing left, including Private Perkins, who is in the middle of the row.

X X X X X X X P X X X X X X X

Any soldier that ends up face to face with another soldier now does a 180° turn. This awkward ritual continues until no soldier can see another soldier’s face.

Can we be sure that Perkins will end up facing the right way?

Aug 21 20

by Brian Gladman

Sunday Times Teaser 3022 – Sporty Set

by Victor Bryant

Published Sunday August 23 2020 (link)

There are 100 members of my sports club where we can play tennis, badminton, squash and table tennis (with table tennis being the least popular). Last week I reported to the secretary the numbers who participate in each of the four sports. The digits used overall in the four numbers were different and not zero.

The secretary wondered how many of the members were keen enough to play all four sports, but of course he was unable to work out that number from the four numbers I had given him. However, he used the four numbers to work out the minimum and the maximum possible numbers playing all four sports. His two answers were two-figure numbers, one being a multiple of the other.

How many played table tennis?