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Oct 26 20

New Scientist Enigma 581 – Trail Blazing Cards

by Brian Gladman

by Keith Austin

From Issue #1734, 15th September 1990

We spent the day camped at the entrance to Apache Pass waiting for the cover of darkness. I passed the time playing cards with a grizzly old prospector, using a pack given him by Billy the Kid.

He took the spades from ace to ten and laid them in a row so:

6 2 9 7 3 10 8 1 4 5

where 1 is the ace.

“That’s the order Wyatt Earp dealt me just before the gunfight at the OK Corral. You can exchange any to cards provided the numbers on them are consecutive but the cards are not next to each other. For example, you can exchange the 7 and the 8, or the 2 and the 3, but not the 4 and the 5”.

“So I can exchange the 7 and 8 and get:

6 2 9 8 3 10 7 1 4 5

and then exchange the 3 and 4 and get:

6 2 9 8 4 10 7 1 3 5

and then exchange the 4 and 5 and get:

6 2 9 8 5 10 7 1 3 4

and so on?” I asked.

“You’ve got it. Now look at this old treasure map I got from Geronimo. It has 6 rows of numbers on it:

A: 7 2 10 9 4 5 3 1 8 6
B: 5 9 2 4 8 1 3 10 7 6
C: 7 4 5 2 1 10 8 3 6 9
D: 4 5 7 8 1 3 2 10 6 9
E: 10 3 4 2 1 9 6 5 7 8
F: 6 10 8 5 3 1 2 4 7 9

“Geronimo said the to discover the treasure it was necessary to find which of these six rows can be obtained from Wyatt’s row by the allowed
operations.”

Find all those of the six rows which can be obtained from Wyatt’s row.

Oct 23 20

Sunday Times Teaser 3031 – End of the Beginning

by Brian Gladman

by Howard Williams

Published Sunday October 25 2020 (link)

Jenny is using her calculator, which accepts the input of numbers of up to ten digits in length, to prepare her lesson plan on large numbers. She can’t understand why the results being shown are smaller than she expected until she realizes that she has entered a number incorrectly.

She has entered the number with its first digit being incorrectly entered as its last digit. The number has been entered with its second digit first, its third digit second etc. and what should have been the first digit entered last. The number she actually entered into her calculator was 25/43rds of what it should have been.

What is the correct number?

Oct 21 20

New Scientist Enigma 579 – The Great Divide

by Brian Gladman

by Susan Denham

From Issue #1732, 1st September 1990

In the two division sums below, the six-figure dividend is the same but the divisor and the answer have been interchanged (that is, the first is x ÷ y = z and the second is x ÷ z = y). No digit occurs in both the divisor and the answer, and no digit in the dividend is used in either the divisor or the answer.

What is the dividend?

Oct 16 20

by Brian Gladman

Sunday Times Teaser 3030 – Pot Success

by John Owen

Sunday October 18 2020 (link)

In snooker, pot success (PS) is the percentage of how many pot attempts have been successful in that match (eg, 19 pots from 40 attempts gives a PS of 47.5). In a recent match, my PS was a whole number at one point. I then potted several balls in a row to finish a frame, after which my improved PS was still a whole number. At the beginning of the next frame, I potted the same number of balls in a row, and my PS was still a whole number. I missed the next pot, my last shot in the match, and, remarkably, my PS was still a whole number.

If I told you how many balls I potted during the match, you would be able to work out those various whole-number PS values.

How many balls did I pot?

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?