western
Physics and Astronomy

                                   
Lightening Calculators
Objective

Lightening Calculators are people who can do mental arithmetic very fast in their heads without using a pencil, paper or a calculator. There are a few very rare individuals who can do this, because they have a special gift, and they are often unaware of how they do the calculations. There are however many strategies that you can learn that will allow you to do mental calculations very quickly, and these are all based on mathematical theorems.

Participants Teams of up to six.
Materials  
Rules

There will be 20 questions on mental arithmetic.
No calculators are allowed.
Nothing may be written except the answer.
There will be 5 seconds per question.
Each question will be stated - then repeated.
Two members of the team will participate in this competition.
You will sit at tables, and the questions will be read out to you.

 

Training for the Event

Here are some concepts that will help.

Formula
Example
Answer
x^2 – y^2 = (x + y)(x – y)
20^2 – 18^2 = (38) (2)
76
(x + y)^2 = x^2 + 2xy + y^2
201^2
= (200 + 1)^2
= 40 000 + 400 + 1

40 401
(x – y)^2 = x^2 – 2xy + y^2

199^2
= (200 - 1)^2
= (40 000 - 400 + 1

39 601
(n + 5)^2 = n ( n + 10 ) + 25

75^2
= (70) (80) + 25
= 5 600 + 25

5 625

Rules for Multiplying

Multiplying by 11
Add ten times the original number to the original number. Example:
248 x 11 = 2480 + 248 = 2728

Rules for Divisibility
Test for 2:
A number is divisible by 2 if the last digit is even. Example:
246 is divisible by 2.

Test for 3:
Sum the digits. If the result is more than 1 digit, sum again. The result is called the digital root of the number. If the digital root is divisible by three, so is the original number.
Example:
246: 2+4+6 = 12; 1+2 = 3, so 246 is divisible by 3.

Test for 4
The number is divisible by 4 if the last 2 numbers are divisible by 4. Example:
248: 48 is divisible by 4, so 248 is so divisible.

Test for 5
The number is divisible by 5 is the last digit is a 0 or 5. Example:
245 is divisible by 5

Test for 6
Test for divisibility by 2 and 3, the factors of 6. The number must be even, and have a digital root divisible by 3. Example:
246 is even, and the digital root is 3, so it is divisible by 6.

Test for 7
There are no simple tests for divisibility by 7.

Test for 8
The number is divisible by 8 if the last three digits are divisible by 8. Example:
1248 is divisible by 8

Test for 9
The number is divisible by 9 if the digital root is also divisible by 9. Example:
252: 2+5+2 = 9, so it is divisible by 9.

Test for 10
The number is divisible by 10 if the last digit is 0. Example:
240 is divisible by 10

Test for 11
Take the number in reverse order, alternately subtracting and adding them. If the result is divisible by 11, so is the original number.
Example:
2948: 8–4+9–2 = 11, so the number is divisible by 11.

Test for 12
The number must be divisible by 3 and by 4.
Example:
252: The digital root is 9, so it is divisible by 3. 52 is divisible by 4. So 252 is divisible by 12.

Judging Each question will be scored out of 5, so that the maximum team score is 75 marks. This score will contribute 33.3% to the overall score for the Mathematics Triathlon. The other components of the Triathlon are Functional Mathematics and Serial Series - Serious Stuff.
Source London District Science Olympics. This event was created by Tom Griffiths.

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