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Puzzles

17 December

If you expand \((a+b+c)^2\), you get \(a^2+b^2+c^2+2ab+2ac+2bc\). This has 6 terms.
How many terms does the expansion of \((a+b+c+d+e+f)^5\) have?

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16 December

Some numbers can be written as the sum of two or more consecutive positive integers, for example:
$$7=3+4$$ $$18=5+6+7$$
Some numbers (for example 4) cannot be written as the sum of two or more consecutive positive integers. What is the smallest three-digit number that cannot be written as the sum of two or more consecutive positive integers?

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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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14 December

The function \(f(x)=ax+b\) (where \(a\) and \(b\) are real constants) satisfies
$$-x^3+2x^2+6x-9\leqslant f(x)\leqslant x^2-2x+3$$
whenever \(0\leqslant x\leqslant3\). What is \(f(200)\)?

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13 December

Today's number is given in this crossnumber. No number in the completed grid starts with 0.

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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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