Puzzles with numbers and things

[davis_greatest]

Here is your half-point, snookersfun

HERE IS THE SCOREBOARD AFTER ROUND 74, WITH POINTS ADDED FOR SOLUTIONS RECEIVED SO FAR TO ROUNDS 68 & 73 WHICH ARE STILL OPEN

snookersfun……………………….…..35½
abextra...............................18½
davis_greatest.....................15½
Vidas..................................12½
elvaago...............................7
chasmmi..............................6½
robert602.............................6
The Statman……………………...……5
Sarmu..................................4
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1

[davis_greatest]

Four - four

[davis_greatest]

Four-nil!

[snookersfun]

OK! I'll give you that. I'll give a little explanation using Fermat's Little Theorem later...

2-0 to Jimmy!

looks like 3-0 now!
I'm calculating my behind off and twisting my brain, and he is telling, 'OK! I'll give you that....'

[davis_greatest]

OK! I'll give you that. I'll give a little explanation using Fermat's Little Theorem later...

2-0 to Jimmy!

[snookersfun]

sorry , 3^#divisible by 9 but not by 6, will give you remainder 6.
for sure 3^987654321 does give remainder 6

[davis_greatest]

Not quite. 3^18, for instance (18 being a multiple of 9) divided by 7 gives remainder 1, not 6.

1-0 to Jimmy against Dott!

[snookersfun]

I don't understand this bit, snookersfun:

amount of times this number is multiplied by itself (for each 9 times it is multiplied by another number with remainder 3, the remainder of the result will always be 6)

any better?

[davis_greatest]

I don't understand this bit, snookersfun:

Multiplying 3, 9, 12, 18... etc. times

[snookersfun]

We have found 2, 3 and 5 as factors.

2 is clearly a factor, since we are adding two odd numbers so must get an even number.

3 is clearly a factor, since 123456789 and 987654321 are both divisible by 3.

And 123456789 to the power of an odd number (123456789) must end in 9; while 987654321 to the power of any whole number must end in 1; so the sum must end in 0. This means that 5 is also a factor (it also shows that 2 is).


I will now tell you that 7 is the 4th smallest prime factor - half a point to anyone who can prove it.

I think, I finally got to the bottom of this one...
I'll try to explain:
123456789 divides by 7 with a remainder of 1. Regardless of the amount of times this number is multiplicated by itself this part of the sum will stay with that remainder.
987654321 divides by 7 leaving a remainder of 3. Multiplying 3, 9, 12, 18... etc. times (and 987654321 is divisible by 9) one obtains the second summand with remainder of 6.
Addition of the two summands (1+6=7) or the final sum is divisible by 7 (no remainder)

Yeah, he comes in handy at times. At least, one of my parents is useful.

[snookersfun]

well done, wish I had a dad like that...

[davis_greatest]

Scoreboard update

Oliver, my pet orang utan, has gone to Barry The Baboon's Ball Shop to buy some snooker supplies. Mainly, Barry sells snooker balls, but he sells other things too.

Oliver buys the following (cheapest first, through to the most expensive):

a packet of chalks
an X to go on the end of a rest
a ball marker
a practice cue ball (with markings to see the effect of spin)


Each item is a whole number of pounds and pence. For example £2.50, or £1.14, or whatever. There are 100 pence in one pound.

Before Oliver pays, being a clever fellow, he works out how much the four items will cost. £7.11 is the total.

Barry, at the till, types the prices in (for example, for £2.50 he would type 2.50). However, Barry makes a mistake, and accidentally presses x (multiply) each time, instead of + (plus).

Oliver, when Barry tells him the "total", never spots a thing, and nor does Barry, because the amount that Barry says is £7.11, as Oliver expected!

How much does each item cost?

Answers by Private Message please. Inital Deadline with be 12:00 noon GMT on Sunday 3 December.

Half a point to austrian_girl's dad, who managed to make Barry's cash register read 7.113942 instead of 7.11!


SO HERE IS THE SCOREBOARD AFTER POINTS HAVE BEEN ADDED FOR SOLUTIONS RECEIVED SO FAR TO ROUNDS 68, 73 and 74, WHICH ARE STILL OPEN

snookersfun……………………….…..35
abextra...............................18½
davis_greatest.....................15½
Vidas..................................12½
elvaago...............................7
chasmmi..............................6½
robert602.............................6
The Statman……………………...……5
Sarmu..................................4
Semih_Sayginer.....................2½
austrian_girl and her dad.........2½
April Madness........................1

[snookersfun]

Attached is a figure, containing a parallelogram ABCD. Also given is that the two angles on C are of same size and ED=AD.

Question: how big is the angle on F (with explanation of course)

Just noticed, I never gave the solutions, here are two possible ways of finding the angle on F, which turned out to be 9O degrees.

Austrian_girl's:
The angle on F is a right angle (90°).

Here comes my explanation. It may not be the simpliest way but anyway: Alle Wege führen nach Rom.
There are two identical angles on C, referred to as 2g (as I can't find a bloody gamma symbol ). This tells us that the angle on D (within parallelogramm) is 180-2g or 2(90-g). Consequently the angle on D (facing triangle) is 180-2(90-g) = 2(90-(90-g)) = 2(90-90+g) = 2g. Ok, this was the baby version! Anyway, I didn't want to leave out a step. Now we know that ED = AD. So both angles at A and E are of identical size. So the angle on E is (180-2g)/2 or simply 90-g. Now, considering the triangle ECF, we have two angles already defined, namely the angle on E (which is 90-g) and the angle on C (which is simply g). As we now, the sum of all angles within a triangle is 180°. So, 180-(90-g)-g = 180-90+g-g = 90°. q.e.d. (hopefully!!!)

Abextra’s:
The angle on F is 90'. As ABCD is a parallelogram, the angles on D and C are equal. As AD and ED are equal, the line that divides the angle on D makes 90' angle(s) with AE. The line is parallel with CF, so the angle on F is also 90'.