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## Math world, simple mental calculations or what’s going on with education?

Today, I want to write blog post which is absolutely not related to programming. It related to math and education in general those days. During work interviews, I see a lot of people, who was absolutely unable to calculate mentally. They just can’t understand, that it’s possible to do without calculators. When my kids (2nd, 6th and 7th grade) were small I taught them to play with numbers, and until 4th grade (bigger kids) they were able do it. but then school teachers “killed” this ability. Why people should use calculator for simple math operations, if he can do it mentally? Shame you, the modern educational system. Let’s go back and try to understand how people were able to live without devil devices, such as calculators…

Following the paint of Nikolai Bogdanov-Belsky “Counting in their heads”. This painting is dated 1895.

As you can see at the painting, peasant kids trying to solve following exercise mentally:

(10

^{2}+ 11^{2}+ 12^{2}+ 13^{2}+ 14^{2}) / 365

This is not very simple exercise, especially when should be solved without your favorite calculator. However, when I was 4th grade I learned to square two-digit numbers mentally (my, and I think, yours too): **First, find the nearest multiple of ten, by raising or lowering your number, then add and remove the rest to each of numbers and add the square of oddment**. For example

45 * 45 = (45+5) * (45-5) + (5 * 5) = 50 * 40 + 25 = (5 * 4) * 100 + 25 = 20 * 100 + 25 = 2000 + 25 = 2025

14 * 14 = (14+4) * (14-4) + (4 * 4) = 18 * 10 + 16 = 180 + 16 = 196

So, now it can be solved easily:

10

^{2}= 100

11^{2}= (11+1) * (11-1) + 1 = 12 * 10 + 1 = 121

12^{2}= (12+2) * (12-2) + 4 = 14 * 10 + 4 = 144

13^{2}= (13+3) * (13-3) + 9 = 16 * 10 + 9 = 169

14^{2}= (14+4) * (14-4) + 16 = 18 * 10 + 16 = 196

And so on… but wait, 100+121+144 already equals 365, which is our denominator. Next sequence will bring us 169+196, which is also 365. So the answer to this black board brain teaser is 2.

However, it can be rather complicated to calculate 86^{2} for instance:

86

^{2}= (86 + 4) * (86 – 4) + (4 * 4) = 90 * 82 + 16…

Let’s try another way – **multiple the difference between the number and 25 by 100, then add the square of the difference or excess of the number and 50**. For example

86

^{2}= (86 – 25) * 100 + (86 – 50)^{2}= 61 * 100 + 36^{2}= 6100 + (36 – 25) * 100 + (50 – 36)^{2}= 6100 + 1100 + 14^{2}= 7200 + 196 = 7396

Isn’t it really simple and fun to calculate squares of numbers?

*Bonus*: how to calculate multiple of two digit numbers with the sum of its unity digits equals to 10?

- Multiply first digit of the first number by 10
- Add 1 to first digit of second number and multiply the result by 10
- Multiply results of step 1 and step 2
- Deduct second number and the result of step 1
- Multiply second digit of the first number by the result of step 4
- Add results of steps 3 and 5

Looks complicated? Let’s make it easier. Assuming that first number is X = 10x + z and second number is Y = 10y + (10 – z), the formula for quick multiplication calculation is: **100 * x * (y + 1) + z * (Y – 10 * x)**. For example:

96 * 84 = 100 * 9 * (8+1) + 6 * (84-10 * 9) = 100 * 9 * 9 + 6 * (84 – 90) = 8100 – 6 * 6 = 8100 – 36 = 8064

37 * 93 = 100 * 3 * (9+1) + 7 * (93 – 10 * 3) = 3000 + 7 * 63 = 3000 + (100 * 6 * 1 + 3 * (7 – 60)) = 3000 + 600 – 3 * 53 = 3600 – 159 = 3441

Have a nice day and be good people. Also, throw out all hardware calculators and uninstall all software

February 8th, 2009 · Comments (8)

## Brilliant, yet simple technical questions, can be used for work interview

Few days ago, a friend of mine, sent me set of demonstration questions for 11th grade computer science students from Russia. I throw eye on this exam and decided to use some of those questions to test candidates in my company for hardware engineer position (we’re looking for). If you’re good candidate for this position, you should be able to answer all those questions without even think a minute.

*Following those questions*:

Automatic system recoded informational message in Russian from Unicode (16-bit) into KOI-8 (8 bit). After recoding the length of the message decreased by 480 bits. What was the length of the original message in characters?

1) 30 2) 60 3) 120 4) 480

There are 119 participants in bicycle race. Special designed system registers cyclers passing waypoint by writing his serial number with minimum possible number of bits. What will be the size of temporary message, recorded by the system after 70 cyclers passed the waypoint?

1) 70 bits 2) 70 bytes 3) 490 bits 4) 119 bytes

**а**=D7_{16}, **b**=331_{8}. What the value of **c** if **a<c<b**?

1) 11011001 2) 11011100 3) 11010111 4) 11011000

Please calculate 43_{8}+56_{16}?

1) 121_{8} 2) 171_{8} 3) 69_{16} 4) 1000001_{2}

What logical expression equals to: & !( !B | C ), while &: logical AND, !: logical NOT and |:logical OR

1) !A | !B | !C 2) A & !B & !C 3) A & B & !C 4) A & !B &C

F is a result of logical expression execution with arguments A, B and C

A | B | C | F |

1 | 0 | 0 | 1 |

0 | 0 | 0 | 1 |

1 | 1 | 1 | 0 |

What is the expression for F?

1) !A & !B & !C 2) A & B & C 3) A | B | C 4) !A | !B | !C

For encoding characters A, B, C and D we decided to use two bits numbers sequence (from 00 to 11). What was the result of coding sequence **BACD** on base 16?

1) 4A 2) 411 3) BACD 4) 1023

Led table contains of number of leds. Each led can be in one of following states: “On”, “Off”, “Blinking”. What is the minimal number of leds should be on table in order to display 18 difference signals?

Please calculate 25_{14} + 11_{-4} – 21_{10}?

To calculator there are only two commands 1: “add 3”, 2: “multiply by 4”. Please write down the sequence of the calculator command in order to convert number **3** to number **57** while using not more, then **6** commands. For example to convert 2 into 50 you should write following: “21211” (X4,+3,X4,+3,+3)

The speed of data transfer over network is 128000 bits/sec. Please calculate the time (in seconds) will take to transfer 625 kBytes in this network.

Write a simple program (by using your favorite programming language) to convert the array of 30 integers into array of modulus of items from input array without using modulus calculation special function.

Good luck!

For people, currently employed in my company (hardware devs) it took less, then10 minutes to solve all those questions. However, I was shocked, that at least half of new candidates, come to interview failed to solve it.

January 30th, 2009 · Comments (26)

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