Image credit: stevepb/ Pixabay
Is the king correct? How many metric tons of rice would be required on the board, assuming they would fit(!), if each grain weighs 0.025g?
(Note: 1,000 kg equals a metric ton.)
TOTAL 1+2+4+8+ ... +2^63 grains of rice
TOTAL GRAINS
=1+2+4+8+…+2^63
Demonstrate the first three lines of a spreadsheet solution. Assuming students know about Filling down, have them complete the task and add a final SUM formula to total all the squares.
(Enter a 1 in A1 and B1
Enter =A1+1 in cell A2 and =B1*2 in cell B2
then Fill down both columns to row 64)
A completed spreadsheet is provided for this lesson sequence (Spreadsheets and iteration worksheet: Chessboard problem).
1.844674407370960000000000000000E+19
(Note: Actual value is
= 18,446,744,073,709,551,615 grains
= 461,168,602,000 metric tons, which would be a pile of rice larger than Mount Everest and around 1,000 times the total global production of rice in 2010.)
The surprising conclusion:
On the entire chessboard there would be
2^64 − 1 = 18,446,744,073,709,551,615
grains of rice, weighing 461,168,602,000 metric tons, which would be a pile of rice larger than Mount Everest and around 1,000 times the total global production of rice in 2010 (which was 464,000,000 metric tons).
End yr 1: $100 x 1.10 = $110.00
End yr 2: $110 x 1.10 = $121.00
End yr 3: $121 x 1.10 = $133.10
Etc
End yr 100: ????
Tell students the answer may surprise them!
These will be the values if row 32 is year 30 (as is the case with the accompanying Spreadsheets and iteration worksheet: Compound interest)
Goal seeking is quite exciting as students see the figures down the spreadsheet rippling as the calculation proceeds – a rare visible example of the time taken for the extensive recalculations involved.
Answer: We need to invest $57,309 now to be a millionaire within 30 years with an interest rate of 10% p.a.
What other iterative problems could we solve using these techniques?
(Moore's law, exponential growth, radioactive decay, Fibonacci problems, factorials)
(The part of the formula using this interest rate will need to have it absolute referenced. It will be instructive for students to fall into this error to amplify the distinction between absolute and relative addressing when filling down a formula.)
(This will likely require the CONCATENATION formula and a column of integers used for this purpose.)
A=P(1+r)^n
=100*(1.1)^100
=$1,378,061.23