Puzzles with numbers and things

[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

[austrian_girl]

[snookersfun]

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

[austrian_girl]

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

[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)

[davis_greatest]

I don't understand this bit, snookersfun:

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

[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]

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]

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

[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]

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]

Four-nil!

[davis_greatest]

Four - four

[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]

Round 75 - Split the cube

Here is an interesting list of numbers. They all have something in common – and are the smallest numbers that have this property. What is it?

153, 370, 371, 407, 165033, 221859, 336700, 336701, 340067, 341067, 407000, 407001, 444664, 487215, 982827, 983221, 166500333, 296584415, 333667000, 333667001, 334000667, 710656413, 828538472

Answers may be posted on this thread.