Can you find the sweet spot? The Wknd Puzzle by Dilip D’Souza

A kind woman gave our Mumbai Harmonics group some laddus. We found two ways to split them, and x=y can help you figure out how many there were. But wait, there’s an even more elegant solution.

This is an easy puzzle really. There’s no doubt you will solve it. But there’s an elegant way to consider it which gives you the solution immediately. Can you find it?

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Our small group of harmonica players, the Mumbai Harmonics, meets at Hanging Gardens on the first Sunday of every month to play for morning walkers. On one recent such Sunday, somebody brought a bag full of besan laddus for us, just to show her appreciation for the music.

Of course we briefly protested this excessive generosity, but truth be told, we all love sweets. So we gratefully accepted and proceeded to divide them. Here’s what we found.

If we handed them all out, we would get four laddus apiece except for one person, who’d get three. On the other hand, if we handed out three apiece, there would be nine laddus left over.

How many laddus did the generous woman bring us?

Hint: A quick bit of algebra solves this. You don’t know how many of us harmonica players there are, so call that number “h” (for “harmonica ustaads”, of course).

If (h - 1) of us get 4 laddus apiece, one of us gets only 3. Thus the total number of laddus is:

4 x (h - 1) + 3

But if all of us get 3 laddus apiece, there are 9 left over. Thus the total also is:

3 x h + 9

So 4 x (h - 1) + 3 = 3 x h + 9.

This tells us that h is 10, meaning there were 10 of us players. And she brought us 39 laddus.

But can you find the elegant, even intuitive way to think this through, with no “h” and no equations?

Scroll down for the answer.

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Answer:

Think of it this way: the woman distributes the laddus as in the first case. Then she takes back one laddu from each harmonica player who got four. That leaves all of us players with three each. Which, you can see immediately, is the second case. Thus the generous lady must have 9 laddus back in her bag, and there are 10 of us. And she came with 39 laddus.