Lucky lead

A quick rabbit sketchboard story. [Characters from Bridge in the Menagerie].


RR is holding xx, AKJTxxx, xx, xx, when he hears his RHO Papa, open 1NT. RR bids 3H (showing a 7 card suit), and his LHO (SB), bids 3NT with the air of someone who has a heart stopper, and it becomes the final contract.

While RR is fiddling with his cards and trying to figure out what his latest lead agreement with HH is, the HJ drops on the table. HH, who had been distracting Papa by taking a sandwich off Papa's plate, points to the HJ as SB spreads the dummy.

This is what Papa sees (lead HJ):

IMPS None	SB ♠ AKx ♥ Qxx ♦ Jxxx ♣ xxx
	Papa ♠ QJx ♥ xx ♦ AKQx ♣ Axxx

W	N	E	S
			1NT
3H	3NT	P	P
P

Papa reasons that RR likely has AJTxxxx or KJTxxxx with RHO holding a singleton honour. As he needs a heart trick to make the contract, the correct play is to play low from dummy.

When the J wins, RR is more surprised than Papa, and proceeds to cash the rest of the heart tricks.

HH remarks that he would have led the HJ too.

Pens and Caps

You have $n$ pens, and $n$ caps for the pens. For every pen, there is exactly one cap that fits. The rest are either loose or tight.

Someone has taken the caps out and jumbled them up. Can you fit the caps on their respective pens, in expected $O(n \log n)$ time?

[Solution]

Kostub's business card problem

This is a problem on Kostub Deshmukh's business card (co-founder of math.chat). The premise of math chat is to do math with friends, and the problem is in that spirit.

I will let the business card speak for itself:

If you are having problems with the image, the problem in text:

In a group of 9 mathematicians, each speaks at most 3 languages, and each pair of mathematicians share a language.

Show that at least 5 mathematicians speak the same language.

Note: I haven't solved it myself yet. Perhaps Kostub will post a solution in the comments sometime.