## Rounding Machine

#### Problem

A particular number machine works as follows.

E.g. 3.4 6.8 6 or 7.9 15.8 15

A different number machine does the following.

E.g. 3.4 3 6 or 7.9 7 14

Notice that 3.4 came out as 6 from both machines, whereas 7.9 came out differently. What must be special about a number for the same value to come out of each machine?

#### Solution

If `x` is the value going into each machine, the machines can be expressed as [2`x`] and 2[`x`] respectively.

All numbers of the form `n.m` under the integer part function will become `n` by definition.

Therefore 2[`n.m`] = 2`n` (i.e. independent of `m`)

But, [2 `n.m`] = [2 (`n` + ^{m}/_{10})] = [2`n` + ^{m}/_{5}]

If `m` 5, 0 `m` 1 [2`n` + ^{m}/_{5}] = 2`n`, whereas for `m` 5, 1 `m` 2 [2`n` + ^{m}/_{5}] = 2`n` + 1.

And so the decimal part of `x` must be less than .5 for 2[`x`] to be equal to [2`x`].

When is [`x` + 0.5] + [`x` 0.5] equal to 2[`x`] and [2`x`]?