## Brainteaser 1542: Precisely what do I mean?

**From The Sunday Times, 29th March 1992** [link]

There is a row of 10 boxes and each box contains one object, which is a knife, a fork or a spoon. Each of the three utensils is in at least one box. Here are five true statements to help you answer the questions below:

- A knife is in more boxes than a spoon is in;
- A spoon is in more boxes than a fork is in;
- A knife is not in precisely five boxes;
- A spoon is not in precisely three boxes;
- A fork is not in precisely half the number of boxes a spoon is in.
How many boxes contain a knife?

How many boxes contain a spoon?

This puzzle was selected for the book *Brainteasers* (2002, edited by Victor Bryant), in which it appeared in the following (slightly different) form:

There is a row of ten boxes and each box contains one object, which is a knife, a fork or a spoon. There is at least one of each utensil. Here are five true statements to help you work out how many of each there are:

- A knife is in more boxes than a spoon is in;
- A spoon is in more boxes than a fork is in;
- A knife is not in precisely five boxes;
- A spoon is not in precisely three boxes;
- A fork is not in precisely half the number of boxes a spoon is in.
How many of each utensil are there?

However I think both these formulations are flawed, in that under any reasonable interpretation there are no solutions. In the comments I present a variation that can be solved.

[teaser1542]

## Jim Randell 11:05 am

on7 April 2019 Permalink |Having read the solution given in the book, I think the spirit of the puzzle can be maintained by phrasing it slightly differently:

And here is my solution:

Suppose there are

Kknives,Fforks andSspoons, each number has a value from 1 to 9.Looking at the statements:

“A knife is in more boxes than a spoon is in”, seems to be an oddly worded way of saying: “there are more knives than spoons”:

Similarly: “A spoon is in more boxes than a fork is in”, seems to be saying:

We then have three statements of the form: “an X is not in precisely N boxes”

I think there are two reasonable interpretations of this:

i.e.:

or:

i.e.:

The following Python 3 program considers these possible interpretations for the last three statements. It runs in 80ms.

Run:[ @repl.it ]Solution:There is 1 fork, 4 spoons, 5 knives.The constraint

(F < S < K)narrows the solution down to 4 possibilities:Only in the case

(F=1, S=4, K=5)do the last three statements all have viable interpretations. These are:or:

We can see each of these is a true statement, but they do not all use the same interpretation of the statement form. (Statement 3 uses the (b) interpretation. Statements 4, 5 use the (a) interpretation). This is why I changed the puzzle wording, so that the statements are made by different people. To have them made by the same person implies a consistent interpretation and gives no viable solutions.

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