## Tuesday, March 24, 2020

### The Funny Calculator

Just like a “normal” calculator,

When you type 1 x 10, it says 10.
When you type 1 + 10, it says 11.

But…

When you type 1 + 1, it says 10.
When you type 10 + 10, it says 100.

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Using this information alone:

How would you count from one to four?

What would the calculator say if you typed in 100 10 ?

Extending the pattern:

Can you extend the pattern to count up to twenty?

What would the calculator say if you typed in 0.1 + 0.1?

What would the calculator say if you typed in 0.1 x 10 ?

What would the calculator say if you typed in 111 + 111 ?

What would the calculator say if you typed in 111 x 10 ?

### Solution is below:

Solutions:

In general, this number system is a base 2 system. The puzzle is more interesting if you’ve never studied the base 2 number system and have to work the system out logically from the statements above. Even if you have studied the base 2 number system, this may give you a new way to think about it.

Using this information alone:

• How would you count from one to four?

1, 10, 11, 100

10 equals two because 1 + 1 = 10

11 equals three because 10 + 1 = 11

100 equals four because 10 + 10 = 100

• What would the calculator say if you typed in 100 10 ?

Four divided by two equals two, so it would say 100 10 = 10, just like a “normal” calculator.

Extending the pattern:

• Can you extend the pattern to count up to twenty?

1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100

• What would the calculator say if you typed in 0.1 + 0.1?

Doubling a number scoots the “decimal” point over one, so

0.1 + 0.1 = 1 and 0.1 has the value of one half.

What would the calculator say if you typed in 0.1 x 10 ?

Multiplying by “10” is doubling in this number system, which means scooting the “decimal” point over once.  So:

0.1 + 0.1   =    0.1 x 10    =     1

• What would the calculator say if you typed in 111 + 111 ?

Doubling a number scoots the decimal point over one, so 111 + 111 = 1110.

• What would the calculator say if you typed in 111 x 10 ?

Multiplying by “10” is doubling in this number system, which means scooting the “decimal” point over once.  So:

111 + 111   =    111 x 10    =     1110

## Saturday, March 21, 2020

### Counting triangles!

How many triangles are there in this picture?

It's not so easy.....

Did you say 7?  If so, you're right!

Watch this interactive video to practice organizing your thoughts when you solve counting problems: