There is a story about a Chinese man who invented the game of Chess. The Chinese emperor at the time said something like ‘what can I give you for your reward?’. The gentleman replied that on his chessboard, he would have one piece of rice on one square, two on the next, four on the next, eight on the next, sixteen on the next and so forth.
I was told this story by my Granddad this afternoon, with the question ‘how many grains of rice were there in total?’. Being the former IT student that I am, I recognised this as being binary – which to a certain extent it is. A square on the chessboard is equivalent to 1 bit, a line on a chessboard is 8 squares, therefore a line is equivalent to a byte and would have 128 grains of rice on the line.
This is where things started to go a bit awry in my mind. Not having any pieces of paper to work things out on, and realising that it wasn’t simply a case of having 8×128 (the ninth square would be 256), I figured that this was impossible for me to work out. Nevertheless, I started to think that I might be able to work it out once I returned home and had the computer in front of me.
8×8 is 64, thought I, therefore a chessboard is effectively 64 bit, and the answer would be that number, multiplied by itself, minus 1 (which in binary would be 11111111111111111111111111111111111111111111111111111111111111111111111111)
Thinking that it would be easy to work out what the 64th number in the binary world would be, I did a quick Google search but gave up. (I have since worked this out, see below.)
That is when I started to write this blog post, thinking that I could allow you all to see me work through this mathematical problem. To give the story a proper retelling, I then searched Google for ‘rice chessboard’ or something like that. What should pop up high on the list, but an article from good old Wikipedia, which not only told me the story – the Chinese gentleman was in fact an ancient Indian mathmetitian named Sessa so -1 point for my grandfather there – but also the answer.
The answer is 18,446,744,073,709,551,615 – which when read out is eighteen million four hundred and forty six thousand seven hundred and forty four million seventy three million seven hundred and nine million five hundred and fifty one thousand six hundred and fifteen (I think anyway).
Now, having the answer to this conundrum has made me a much better person. Admittedly, I didn’t work it out myself (which actually would have made me a much better person) because I looked it up on the Internet. I was going to, but the Internet stopped me from doing it as it made it quite a pointless thing to do. The Internet has ruined my life. But it has made me a better person for it.
Possibly.
(To conclude, the 64th number in the ‘binary sequence’ is √(18446744073709551615 + 1) which is 4294967296. That bit was worked out by myself, so I wouldn’t be surprised if it’s wrong…)
hehe, thats briliant, however, i found it easier to say that is is 2 to the power of 64 (the number of chess squares, raised to the binary seqence, which is obviously 2, (0,1))
= 18 446 744 073 709 551 616 then take the square root
which is = 4 294 967 296.
therfore you are right, how ever the numbers would be read;
eighteen-pentillion, four-hundred and forty-six-quadrillion, seven-hundred and fourty-four-trillion, seventy-three-billion, seven-hundred and nine-million, five-hundred and fifty-one-thousand, six-hundred and six-teen.
this way is the most common way of saying the number, however Richie and I know you this is the american way so for you, (and me) in proper english would be read;
eighteen-trillion, four-hundred and forty-six-thousand, seven-hundred and forty-four-billion, seventy-three-thousand seven-hundred and nine-million, five-hundred and fifty-one-thousand, six-hundred and six-teen.
therfore you are right.
Also for compleateness, the square number is read;
four-billion, two-hundred and ninty-four-million, nine-hundred and sixty-seven-thousand, two-hundred, and ninty-six. and in proper english;
four-thousand-two-hundred and ninty-four-million, nine-hundred and sixty-seven-thousand, two-hundred, and ninty-six.
try this for fun I have the answer, and i dont mind if you use the internet to help you.
Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black. The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag. A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube. What is the probability that the outside of this cube is completely black?
Also how many squares are on a chess board and what is the pattern within the answer?
try this without the aid of the internet, cos i did this in primary school and am sure you’ll get it fairly easily!!
sorry about all the maths questions i just like them, there fun, hehe.
Well, as mentioned above you will find that there are 64 squares on a chess board. I’m not entirely sure what you mean by pattern within the answer. Obviously it’s an 8×8 square which are half white half black (32 of each). If you want to put items on the board without any on the diagonal or the straight lines you use the move that the Knight (or ‘horsey’ if you will) uses whilst playing Chess.
Other than that, I don’t know what patterns you might be referring to…
no its much deeper than that, here’s a hint the 8×8 square is a square it self therefore there is another the answer is 204, how did i come up wkth this answer and when you figure it out there is a deffinate pattern tha emmerges. 8×8 and 1×1 are a good start no fill in the gaps.
Ah I get it – so you have the 1×1 squares, 2×2 squares, 3×3 squares etc.
There are, of course, 8×8=64 available 1×1 square squares, working right down to 2×2=4 available 7×7 square squares on the board, and of course the 1 (or 1×1=1 for completeness) 8×8 square.
So, you have the square numbers:
1×1=1
2×2=4
3×3=9
4×4=16
5×5=25
6×6=36
7×7=42
8×8=64
Adding them all up, you have
1+4+9+16+25+36+42+64 = 197
Ah, I’m missing some somewhere if you got 204…
OK, I worked out (ish) where I went wrong…
7×7 is 49 not 42…
Oh, how foolish of me…