Eating Biscuit Puzzle

Question

Every week the Hatter bought eight packets of twenty biscuits each. (Each packet contained twenty biscuits.) He would devour these with such carelessness, however, that he could never eat an entire biscuit: there would inevitably be an end left over from each biscuit.

To try to save money, he kept these uneaten ends to squish into more biscuits. From four ends, rather cunningly, he could make one entirely new biscuit. How many biscuits did the Hatter eat a week, including those made up from the ends of others?

Note: Your answer should be an integer, ignoring the last uneaten ends which were not enough to make a new biscuit.

I got this question from a 6th-grader. To be honest, I don't quiet understand this question, as it's not really specifying how many biscuits the Hatter is eating each day. From my understanding, the Hatter is eating 3/4 of each of 160 biscuits. So there are 40 biscuits made of the uneaten ends each week?

Thanks in advance

Answer 1

Note: Your answer should be an integer, ignoring the last uneaten ends which were not enough to make a new biscuit.

He eats $(160)$ biscuits, leaving $160$ ends.

He combines these into $40$ more biscuits and eats them, leaving $40$ biscuit ends.

He combines these into $10$ more biscuits, and eats them, leaving $10$ biscuit ends.

He combines these into $2$ more biscuits, with $2$ biscuit ends left over, and eats them.

At this point, there are $2$ biscuit ends left over from the $10$ biscuit ends, and another $2$ biscuit ends from the $2$ biscuits just eaten.

So, $1$ more biscuit is eaten.

So, the number of biscuits eaten is

$$160 + 40 + 10 + 2 + 1 = 213.$$

Answer 2

If he's buying 160 biscuits/week, then ultimately he's consuming 160 biscuits/week. But if "eating a biscuit" means consuming 3/4 of a biscuit, then he's "eating a biscuit" $160\times\frac 43$ times/week. Since he discards uneaten pieces at the end of each week, we round this down to the nearest integer to get 213.

Answer 3

This is an old question, which used to be asked in terms of cigarettes when I typically saw it as a kid.

In terms of biscuits, the Hatter first eats $160$ biscuits, leaving $160$ ends. These can be combined to make $160/4 = 40$ whole biscuits, which the Hatter eats, now leaving $40$ ends. These in turn can be combined to make $40/4 = 10$ whole biscuits, which the Hatter once more eats, leaving $10$ ends. These can now be combined to make $2$ biscuits, with $2$ ends left over for the moment.

But now the Hatter eats these two biscuits, leaving $2$ ends. He combines them with the $2$ ends that were previously left over, making one last whole biscuit. He eats this biscuit, leaving a single end, which cannot be combined anymore. (Although I must say I don't understand why he doesn't just eat it.)

In all, the Hatter eats $160+40+10+2+1 = 213$ "biscuits."