Which is greater:

the 789th number in the 123 times table to begin with the digits 456,

or the 123rd number in the 789 times table to begin with the digits 456?

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Which is greater:

the 789th number in the 123 times table to begin with the digits 456,

or the 123rd number in the 789 times table to begin with the digits 456?

An expedition is being planned to climb a truncated conical volcano that is 40km diameter at the base, and 24km diameter at the crater edge. The slope length is 12km.

The path starts at A at the base and ends at point B on the crater edge, exactly halfway round the volcano from A.

What is the length of the shortest possible route around the conical face of the volcano (to the nearest km)?

Bakewell parkrun is a picturesque free, weekly, timed 5km run along the Monsal trail in Derbyshire. The course heads off 2.5km along the trail, then takes a u-turn, before heading back to the start/finish.

Two friends run the parkrun, both starting at the same time (Saturday 9am). They run at different paces, but their paces are consistent. Nicola, the faster of the two, reaches the turnaround point first, then exactly two minutes and twenty seconds later, passes Danny travelling the other way. At the finish, Nicola has exactly twelve minutes to wait before Danny crosses the finish line.

How long do Nicola and Danny take to complete their parkruns?

Runners will know, if they want to predict what time they should be aiming for in, say, a 10km race, based on what they can do at 5km, you should do more than merely doubling the time: the longer a race, generally the slower you need to go. One useful rule is known as the 6% rule. It isn’t unfortunately as simple as merely doubling the 5km time and adding 6% - instead it is raising the ratio of distances to a power of 1.06.

So (10/5)^1.06, would give you a factor of around 2.085. Multiply this by your typical 5km time and you should be looking at that for your 10km time.

Of course, this is only a rule of thumb, so will be more true for some people than for others. For one athlete, let’s call him Norm, he finds that his personal best times, at a variety of distances, precisely follow the rule (to the nearest second). His half-marathon best time is exactly 1 hour more than his 10km best time. (A half-marathon in metres is 21097.5).

What are his 10km and half-marathon personal bests?

This conversation took place on a particular day of the (non-leap) year:

“Isn’t it curious,” remarked Izzie, “that if you take today’s date, double the date number and double the month number, you get my birthday?”

“Interesting!” replied Leila, “but if you consider today as the ‘n’th day of the year and work out what the ‘2n’th day of the year is, you get MY birthday!”

Izzie’s and Leila’s birthdays are in the same calendar month.

When did this conversation take place and when are the girls’ birthdays?

Can you add two more bars to this digital display to make it a multiple of 13 that is NOT 78?

Place each of the numbers 1 to 20 in a row, in such an order that the sum of any pair of adjacent numbers is a number that appears in the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

In a betting game there are five cards placed face down. Each has a number on it.

You win the bet if you end up with the card with the highest number.

You can select any of the cards at random, and having seen the number you can decide whether to accept the card or reject it and choose another, and you may do this as many times as you like, however once you have rejected a card you cannot get it back.

The actual values on the card give you no indication of whether you might have the highest, for instance ‘1’ may be the highest number, or ’50,000,000’ may be the lowest.

Since there are five cards you might think a fair price for the bet would be 4 to 1 against (you place £1 against the house’s £4 and if you win you take all £5). Instead you are offered a measly 3 to 2 against (you place £2 against the house’s £3 and if you win you take all £5).

Is it possible to beat the house? If so how?

There are seven circles of various sizes, each touching its neighbours.

Connecting the centres of the circles creates a heptagon. The edges of the heptagon are given, except for one.

The only other information we have is that the two bold circles are the same radius.

What is the missing length?

I recently stumbled upon an intriguing theorem that stated that every positive whole number can be expressed as the sum of three palindromic numbers, although it is often far from straightforward to find them.

Can you express the number 6878592 as the sum of three palindromes?

It is the sum of three numbers of 7, 6 and 5 digits respectively, and no digit is repeated except where being a palindrome dictates it:

ABCDCBA + EFGGFE + HJKJH = 6878592

We have a whole number, let’s call it ‘b’.

We also have a number of statements:

1 divides exactly into ‘b’

2 divides exactly into ‘b’

3 divides exactly into ‘b’

4 divides exactly into ‘b’

… continuing all the way to …

‘a’ divides exactly into ‘b’

We are told that three consecutive statements in the list are NOT true, while all of the others are true.

Question a: what is the MAXIMUM value that ‘a’ could possibly be?

Question b: given that value for ‘a’, what is the MINIMUM value that ‘b’ could possibly be?

An industrial process needs 7kg of material to produce one widget, but in doing so produces 2kg of waste material which can then go towards making another widget.

If I start with 4937kg of material, how many widgets can I make altogether?

I celebrated my birthday this week. Right now, my age, and those of my wife, my daughter and my son, are all prime numbers. After my daughter celebrates her next birthday in a few months we will not all be prime ages until 30 years from now. And after that (if we are all still around) it would be another 30 years.

My daughter is about four and a half years older than my son, and I am about one and a half years older than my wife. How old are we all?

I have a loop of string N centimetres long. I can form this loop into a square with n cm along each side (n being exactly a quarter of N). Not surprisingly I find that if I draw diagonals from opposite corners of this square, they cross at right angles. I also find that I can tilt the square into a rhombus shape, while still having n cm along each side, the diagonals still cross at right angles. Still not surprising.

However, I discover that if I move some of the corners so that instead of all being n cm long, the four edges of the quadrilateral are n, n-5, n+1 and n+4, then the diagonals STILL cross at right angles.

How long in my loop of string?

Replace each letter in this code either by the letter preceding it or following it in the alphabet.

There is a rickety old carousel, with 24 horses equally spaced around the edge of a disc. There are at least two things wrong with it: one is that the horses in positions A, B, C, D, E, F and G are broken and not safe to ride; another is that the carousel needs to be perfectly balanced around the centre of the disc, otherwise it won’t go round. This means that when there are two children wishing to ride, they would have to be positioned in diametrically opposite positions, eg, A and M, in order that the centre of gravity is at the centre of the disc. If there were three children they could be at positions A, I and Q.

On this particular day, nine children wish to ride the carousel. Where do they need to be positioned such that the centre of gravity is at the centre of the disc, and that positions A to G are unoccupied?

Find a four digit number ABCD such that ABCD x ABC x AB x A = 369,457,740