A hotelier wanted to furnish the dining room with new chairs. In the catalog, he chose a type of chair. Only when placing the order did he learn from the manufacturer that as part of a discount offer, every fourth chair was offered at half price, and that he could thus save the equivalent of seven and a half chairs compared to his original plan. The hotelier calculated that with the originally planned amount, he could purchase nine more chairs than he intended.

How many chairs did the hotelier originally want to buy?

(L. Simünek)

Hint. First solve the problem without the information that with the originally planned amount, nine more chairs can be purchased.
In the promotion, every fourth chair was at half price. If the hotelier could save the equivalent of 7.5 chairs, he ordered 15 sets of four chairs and at most three additional chairs, i.e., he ordered at least 60 and at most 63 chairs.

Compared to the original plan, he could afford 9 more chairs during the promotion, i.e., at least 69 and at most 72 chairs. In the first case, $69=17 \cdot 4+1$, so the savings would amount to only $17 \cdot \frac{1}{2}=$ $=8.5$ chairs, which does not match the assumption. The same conclusion applies to the next two possibilities: $70=17 \cdot 4+2$ and $71=17 \cdot 4+3$. Only for $72=18 \cdot 4$ does the savings amount to exactly 9 chairs.

Of the four considered possibilities, only the last one is the solution to the problem. The hotelier originally intended to buy 63 chairs.

Another solution. Similarly to the previous solution, we determine that the savings of 7.5 chairs correspond to an order of at least 60 and at most 63 chairs $\left(7.5=\frac{1}{2} \cdot 15\right.$ and $\left.15 \cdot 4=60\right)$.

Similarly, we determine that the savings of 9 chairs correspond to an order of at least 72 and at most 75 chairs $\left(9=\frac{1}{2} \cdot 18\right.$ and $\left.18 \cdot 4=72\right)$. This number should be 9 more than the number of chairs in the previous order, so the savings of 9 chairs should correspond to an order of at least 69 and at most 72 chairs. These two conditions are met only in one case, which corresponds to 72 in the second, and 63 in the first order.

The hotelier originally intended to buy 63 chairs.