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Puzzles

24 December

When written in binary, the number 235 is 11101011. This binary representation starts and ends with 1 and does not contain two 0s in a row.
What is the smallest three-digit number whose binary representation starts and ends with 1 and does not contain two 0s in a row?

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21 December

There are 6 two-digit numbers whose digits are all 1, 2, or 3 and whose second digit onwards are all less than or equal to the previous digit:
How many 20-digit numbers are there whose digits are all 1, 2, or 3 and whose second digit onwards are all less than or equal to the previous digit?

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19 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.
+= 7
× × ×
+= 0
÷ ÷ ÷
+= 2
=
4
=
35
=
18

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Tags: numbers, grids

18 December

Some numbers can be written as the product of two or more consecutive integers, for example:
$$6=2\times3$$ $$840=4\times5\times6\times7$$
What is the smallest three-digit number that can be written as the product of two or more consecutive integers?

15 December

The arithmetic mean of a set of \(n\) numbers is computed by adding up all the numbers, then dividing the result by \(n\). The geometric mean of a set of \(n\) numbers is computed by multiplying all the numbers together, then taking the \(n\)th root of the result.
The arithmetic mean of the digits of the number 132 is \(\tfrac13(1+3+2)=2\). The geometric mean of the digits of the number 139 is \(\sqrt[3]{1\times3\times9}\)=3.
What is the smallest three-digit number whose first digit is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers?

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12 December

What is the smallest value of \(n\) such that
$$\frac{500!\times499!\times498!\times\dots\times1!}{n!}$$
is a square number?

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11 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.
++= 15
+ + ÷
+= 10
+ ×
÷×= 3
=
16
=
1
=
30

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Tags: numbers, grids

10 December

How many integers are there between 100 and 1000 whose digits add up to an even number?

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