Here is a number trick that is not too well known and so you can use it to impress your friends. In fact this doesn’t require a computer and can even be done with paper and pencil. It involves a series known as the Fibonacci number sequence, named after the Italian mathematician Leonardo Fibonacci (1170-1250).
A Fibonacci number sequence is formed by starting with any two numbers, adding those to get a third number, adding the second and third to produce a fourth number and so on.
This is much easier to see with a short example:
2
3
5
8
13
21
34
55
89
144
As can be seen, the sequence is formed by adding the previous two numbers.
2 plus 3 = 5, 3 plus 5 = 8, 5 plus 8 = 13, 8 plus 13 = 21 and so on.
Using the calculator below, if you input 2 and 3 into the first two boxes, when you click “Calculate”, you will see all 10 boxes filled in with the same numbers in the list above.
Now, for the “trick” with the Fibonacci number sequence.
Ask your friend for two numbers.
You could then enter the numbers in this computer page but it is much more impressive if this trick is done on paper.
Adding the two numbers, create a Fibonacci sequence that is exactly ten steps long.
When you reach the tenth number, tell your friend you can total all ten numbers in your head!
And what’s the secret?
Whenever you have a Fibonacci sequence of 10 numbers, the total will always be the seventh number times 11.
For practice, input two and three in the first two boxes and then click “CALCULATE”.
Yes, you could get the total by clicking “Calc Total” or by using the trick.
If you entered two and three for the first two numbers, the seventh number will be 34 and multiplying this by 11 gives a result of 374.
It isn’t that difficult to multiply by 11 in your head.
For example, to multiply 34 by 11, think of summing 34 and 34 but shift one number one decimal place.
34 34 374
Obviously, when you ask a friend for two numbers make sure you mention they should be kept relatively small.
Doing this trick on paper is much more impressive than on a computer so why not use this computer page just for practice?
Remember, before you show this to anybody, the best advice is to practice.
And, if you are curious, here are the first 100 Fibonacci numbers:
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
| 8 | 21 |
| 9 | 34 |
| 10 | 55 |
| 11 | 89 |
| 12 | 144 |
| 13 | 233 |
| 14 | 377 |
| 15 | 610 |
| 16 | 987 |
| 17 | 1,597 |
| 18 | 2,584 |
| 19 | 4,181 |
| 20 | 6,765 |
| 21 | 10,946 |
| 22 | 17,711 |
| 23 | 28,657 |
| 24 | 46,368 |
| 25 | 75,025 |
| 26 | 121,393 |
| 27 | 196,418 |
| 28 | 317,811 |
| 29 | 514,229 |
| 30 | 832,040 |
| 31 | 1,346,269 |
| 32 | 2,178,309 |
| 33 | 3,524,578 |
| 34 | 5,702,887 |
| 35 | 9,227,465 |
| 36 | 14,930,352 |
| 37 | 24,157,817 |
| 38 | 39,088,169 |
| 39 | 63,245,986 |
| 40 | 102,334,155 |
| 41 | 165,580,141 |
| 42 | 267,914,296 |
| 43 | 433,494,437 |
| 44 | 701,408,733 |
| 45 | 1,134,903,170 |
| 46 | 1,836,311,903 |
| 47 | 2,971,215,073 |
| 48 | 4,807,526,976 |
| 49 | 7,778,742,049 |
| 50 | 12,586,269,025 |
| 51 | 20,365,011,074 |
| 52 | 32,951,280,099 |
| 53 | 53,316,291,173 |
| 54 | 86,267,571,272 |
| 55 | 139,583,862,445 |
| 56 | 225,851,433,717 |
| 57 | 365,435,296,162 |
| 58 | 591,286,729,879 |
| 59 | 956,722,026,041 |
| 60 | 1,548,008,755,920 |
| 61 | 2,504,730,781,961 |
| 62 | 4,052,739,537,881 |
| 63 | 6,557,470,319,842 |
| 64 | 10,610,209,857,723 |
| 65 | 17,167,680,177,565 |
| 66 | 27,777,890,035,288 |
| 67 | 44,945,570,212,853 |
| 68 | 72,723,460,248,141 |
| 69 | 117,669,030,460,994 |
| 70 | 190,392,490,709,135 |
| 71 | 308,061,521,170,129 |
| 72 | 498,454,011,879,264 |
| 73 | 806,515,533,049,393 |
| 74 | 1,304,969,544,928,657 |
| 75 | 2,111,485,077,978,050 |
| 76 | 3,416,454,622,906,707 |
| 77 | 5,527,939,700,884,757 |
| 78 | 8,944,394,323,791,464 |
| 79 | 14,472,334,024,676,221 |
| 80 | 23,416,728,348,467,685 |
| 81 | 37,889,062,373,143,906 |
| 82 | 61,305,790,721,611,591 |
| 83 | 99,194,853,094,755,497 |
| 84 | 160,500,643,816,367,088 |
| 85 | 259,695,496,911,122,585 |
| 86 | 420,196,140,727,489,673 |
| 87 | 679,891,637,638,612,258 |
| 88 | 1,100,087,778,366,101,931 |
| 89 | 1,779,979,416,004,714,189 |
| 90 | 2,880,067,194,370,816,120 |
| 91 | 4,660,046,610,375,530,309 |
| 92 | 7,540,113,804,746,346,429 |
| 93 | 12,200,160,415,121,876,738 |
| 94 | 19,740,274,219,868,223,167 |
| 95 | 31,940,434,634,990,099,905 |
| 96 | 51,680,708,854,858,323,072 |
| 97 | 83,621,143,489,848,422,977 |
| 98 | 135,301,852,344,706,746,049 |
| 99 | 218,922,995,834,555,169,026 |
| 100 | 354,224,848,179,261,915,075 |
An interesting aspect of the Fibonacci Number Sequence is that if you divide one Fibonacci number by the previous Fibonacci number, this produces a quotient called the phi ratio φ which is also known as the golden ratio.
For example, Fibonacci #20 divided by Fibonacci #19 =
Another way to express the phi ratio is:
φ = 0.5 + 5.5 • 0.5 = 1.6180339887498948482…
The Fibonacci Sequence appears in many places.
Here is a website that explains the arrangement of seeds in a sunflower being based on the Fibonacci Sequence.
Leonardo Fibonacci (1170 – 1250) is thought to be the western world’s most skillful mathematician of the Middle Ages.
In his 1202 book Libre Abaci he strongly advocated for the use of Arabic Numerals as opposed to Roman Numerals.
He stated that Arabic Numerals were much easier to read and calculations could be done more quickly and accurately with them.