Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 163
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The hostess has bought a cake at the shop. She does not eat sweets herself, but she has 7, 8 or 9 guests coming. She has to cut the cake in advance. With what is the smallest number of pieces she can divide the cake equally among the guests in all three cases?
I should say right away that the answer 504 is not correct, although it seems obvious at first sight, because it is the smallest common multiple of 7,8 and 9. In short, you can do it in fewer chunks.
By the way, the pieces can also be of different sizes.
The piece is first cut into 7 pieces. Then cut each of the 7 pieces into pieces.
The proportions are as follows 18 pieces at 1/1008, 18 pieces at 1/1296 and 1 piece at 1/9. In total you get: 18*1/1008+18*1/1296+ 1*1/9= 1/7
Then we do the following:
- if there are 7 left, we give them each {18 parts of 1/1008, 18 parts of 1/1296 and 1 part of 1/9} = 1/7
- If 8 are left, 7-ry will be given 18 pieces of 1/1296 and 1 piece of 1/9, so 18/1296+1/9=1/8,
and one gives the rest, i.e. 7*18=126 pieces of 1/1008=126/1008=1/8.
- If there are 9 people left, then 7 people get 1/9 of a piece each,
and we divide the remainder by the remaining two. The remainder we have is 7*18 parts of 1/1008 and 7*18 parts of 1/1296
or 126 parts of 1/1008 and 126 parts of 1/1296.
Dividing it equally we get: 63 parts of 1/1008 and 63 parts of 1/1296, i.e. 63/1008+63/1296=1/9
=====================================
So we have a total of 7*(18+18+1)=259parts.
=====================================
Check: 7*1/9+7*18*1/1008+7*18*1/1296=1 (whole brick of gold).
A more radical option:
Cut the piece first into seven pieces. Then cut each of the 7 pieces into pieces.
The proportions are as follows: 2 parts 1/112, 2 parts 1/144 and 1 part 1/9. The total is: 2*1/112+2*1/144+ 1*1/9= 1/7
Then we do the following:
- if there are 7 left, we give each {2 parts 1/112, 2 parts 1/144 and 1 part 1/9}=1/7
- if there are 8 left, 7-ry will be given 2 pieces of 1/144 and 1 piece of 1/9, i.e. 2/144+1/9=1/8
and one will get the rest, i.e. 7*2=14 pieces of 1/112=14/112=1/8.
- If there are 9 people left, then 7 people get 1/9 each,
and we divide the remainder by the remaining two. The remainder we have is 7*2 parts 1/112 and 7*2 parts 1/144
or 14 parts of 1/112 and 14 parts of 1/144.
Dividing it equally we get: 7 parts 1/112 and 7 parts 1/144, i.e. 7/112+7/144=1/9
============================
So we have a total of 7*(2+2+1)=35parts.
============================
Check: 7*1/9+7*2*1/112+7*2*1/144=1 (whole brick of gold).
Do you have the right solution yourself? The problem is very recent...
First cut a piece into 7 pieces. Then cut each of the 7 pieces into pieces.
The proportions are as follows 18 pieces at 1/1008, 18 pieces at 1/1296 and 1 piece at 1/9. In total you get: 18*1/1008+18*1/1296+ 1*1/9= 1/7
Then we do the following:
- if there are 7 left, we give them each {18 parts of 1/1008, 18 parts of 1/1296 and 1 part of 1/9} = 1/7
- If 8 are left, 7-ry will be given 18 pieces of 1/1296 and 1 piece of 1/9, so 18/1296+1/9=1/8,
and one gives the rest, i.e. 7*18=126 pieces of 1/1008=126/1008=1/8.
- If there are 9 people left, then 7 people get 1/9 of a piece each,
and we divide the remainder by the remaining two. The remainder we have is 7*18 parts of 1/1008 and 7*18 parts of 1/1296
or 126 parts of 1/1008 and 126 parts of 1/1296.
Dividing it equally we get: 63 parts of 1/1008 and 63 parts of 1/1296, i.e. 63/1008+63/1296=1/9
=====================================
So we have a total of 7*(18+18+1)=259parts.
=====================================
Check: 7*1/9+7*18*1/1008+7*18*1/1296=1 (whole brick of gold).
You forgot about the option with the five remaining.
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I've got 60 pieces to divide:
5 pieces at 1 / 9
10 pieces at 1 / ( 9 * 5 )
5 pieces at 1 / ( 9 * 7 )
10 pieces at 1 / ( 9 * 7 * 5 )
5 pieces at 1 / ( 9 * 8 )
10 pieces at 1 / ( 9 * 8 * 5 )
5 pieces at 1 / ( 9 * 8 * 7 )
10 pieces at 1 / ( 9 * 8 * 7 * 5 )
You forgot about the option with the five remaining.
How is that five if there are only three choices - seven, eight or nine?
Well, yes. Pardon me. I didn't read the assignment carefully.
That makes 28 pieces.
The wording of the problem has been changed at the request of one of the moderators of the Megamoscow site for the purpose of anti-googling.
The essence of the problem does not change.
I apologise to the authors of the posts affected.
5 pieces for 1 / 9
10 pieces at 1 / ( 9 * 5 )
5 pieces at 1 / ( 9 * 7 )
10 pieces at 1 / ( 9 * 7 * 5 )
5 pieces at 1 / ( 9 * 8 )
10 pieces at 1 / ( 9 * 8 * 5 )
5 pieces at 1 / ( 9 * 8 * 7 )
10 pieces at 1 / ( 9 * 8 * 7 * 5 )
This can be done, imagine, for some 22 pieces :) But probably won't show it yet, if you don't mind.
As it turns out, you can do even less, but my solution for now is 22.
Thank you, but it's too complicated for me.
As it turns out, even less is possible, but my solution so far is 22.