Math for challenging brains

here are few mathematical puzzles to challenge your bdrain

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1. In a ten-digit number, first represents number of “1” present in the number, second digit  represents no of “2” in that number and third digit represents no of “3” in that number and so on ninth digit represents no of nines in that number and tenth digit represents no of “0” in that number. Then what is that ten digit number?

We start with the smallest possible ten digit number is 1000000009. Now, since last digit represents number of zeros, it should be 9.

Hence, the number becomes 1000000009.

Since, we have only 8 zeros so the last digit should be 8. Hence, the number becomes 1000000008.

Since, number of “8” present in the above case is one, next modification will be 1000000108.

Now, the number of “0” is only seven. Hence, the

number is modified to 1000000107.

Since, the number of “7” present is one so next modification will be 10000001007.

Now, since the number of “1” present is two, the next modification will be 2000001007.

Number of “2” present is to be one so the next modification will be 2100001007. 

Here, since number of zeros present is six, the next modification will be 2100001006. 

Now, since number of “6” present is 1, so the next modification will be 2100010006. 

The number 2100010006 satisfies all the condition so this is the desired result.

2.without actually calculating x, show that \\ \text{If } x+1/x=1,\enskip \text{then prove } x^7+1/x^7=1
\text{Since } x+1/x=1, \\\text{then multiplying through by x gives } x^2+1=x \text{ or, rearranging }, x^2=x-1 \\ \text{Multiplying through by x again,} \\ x^3=x^2-x \\ \text{or, substituting our expression for } \\ x^2, x^3=x-1-x=-1 \\ \text{Squaring this gives }\\ x^6=1,\\ \text{and then multiplying through by x we have } \\ x^7=x \\ \text{So, substituting }x^7 \text{ for x in the original } \\ x+1/x=1 \\ \text{we have } \\ x^7+1/x^7=1 \\ \text {and we are done}.

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