Textbook 6H

This is a hand from the intra google bridge tournament (a global tournament among Google employees).

You are South holding -,JT8xxxx, AQx, AKx and open 1H after RHO passes as dealer. You hear partner bid 3H (10-12 limit raise with 4).  What will you bid?

Say you just somehow end up in 6H.


LHO leads the CT and you see:

IMPS E/W	Dummy ♠ AT32 ♥ A974 ♦ 865 ♣ QJ
	You ♠ - ♥ JT86532 ♦ AQ7 ♣ AK2

Contract:6H

First Lead: ♣T

How will you play?










The problem will be if hearts don't split and DK is offside. You can cater to some of that via an end play.

Win the club in dummy, cash SA throwing a diamond, and ruff a spade (key play).

Now play a heart to the A. If trumps divide, you can take the diamond finesse for the overtrick. If RHO has the KQ, you have to rely on the diamond finesse.

If LHO has the trump KQ (as it was at the table) now you can still practically guarantee your contract. Ruff a spade to hand, cash the clubs throwing a spade and exit a heart. Now LHO has to play a diamond or give a ruffnsluff.

As fate would have it, LHO had the DK too, so this was a required play to make the slam.

This hand was bid and made by Wei-Bung Wang (who has played internationally for the Taiwain junior national team). The other table was only in 4H so making it was a huge swing (as compared to going down).

Wei-Bung bid 6H directly after the 3H and this is his reasoning:

I opened second hand. RHO didn’t open 2S, LHO didn’t overcall 1S. Partner rates to have some spades. If he has 5-4-2-2 and no strength at all, the slam is still 26%. There’s no scientific way to stop at 4-level. There’s also no scientific way to reach grand slam.

Defensive 4H

In an IMP team game, you are East and hold x, AQx, QJxxx, Kxxx

(If you need to know what an x is, assume lowest spots).

Your partner is dealer and opens 2S , RHO bids 3C, you pass, LHO bids 4H which ends the auction.

Partner leads the SK and yoy see:

IMPS None	Dummy ♠ Axx ♥ x ♦ Axxx ♣ AQJ9x
		You ♠ x ♥ AQx ♦ QJxxx ♣ Kxxx

W	N	E	S
2S	3C	P	4H
P	P	P

Declarer wins the SA, cashes DA throwing a spade and plays a heart.

What is your plan?









If you go up with the HA, you get 2 hearts and a club, but that is all. Declarer can easily discard the last spade loser on clubs.


You must hope partner has Jx or Tx of hearts and play the Q!

Imagine you are declarer with KJ9xxxx  and see the Q show up. You could try winning the K and play low to cater to AQ with RHO. If declarer ducks and it is indeed AQ tight, RHO could maneuver a club ruff for partner.

Declarer could still get it right, but has to guess. If declarer guessess wrong, partner will get in with a heart to cash his spade. You now get 2 hearts, 1 spade and 1 club to beat the contract. If declarer guesses right you have just let them make an overtrick.


At MPs this is harder and going up with the A to guarantee the second heart trick is probably the percentage play.

Integer polynomial property

$P$ is a polynomial with integer coefficients. Show that if $a$ is an integer such that

$$ P(P(P(a))) = a$$

then

$$ P(a) = a$$


Solution Sketch:


This uses the fact that $P(x) - P(y)$ is divisible by $x - y$ to get a cyclic chain of divisibility conditions implying each one in the chain is $\pm1$ times the others. Some assumptions like $P(a) \gt a$ etc lead to contradictions.

First trick decision in 3H (and defensive tidbit too)

This was MP, but assume IMPS.

You are South and end up in 3H. LHO leads CQ.

IMPS None	North ♠ Kx ♥ AKx ♦ KQT98 ♣ Kxx
	South ♠ xxx ♥ QJTxx ♦ Jx ♣ xxx

W	N	E	S
1S	X	P	2H
P	3H	P	P
P

What is your plan? Do you cover or duck the CQ? Do you think you can make this?

At the table I ducked the CQ and the CJ continuation which held too.

LHO now shifted to a heart and it was all over. I could draw trumps and play on diamonds with a spade entry to dummy.

After the CJ held, LHO who held AJTxxx, x, Axxx, QJ could shift to the SJ (or T) to knock out the SK, then use a low spade to get to partner's hand to cash the setting club trick! RHO could have overtaken the CJ to give partner a ruff but that is losing defense when partner has QJx.

A simpler defense by West is to just play SA and another but that might not work against a hand like xx, Qxxxxx, Jx, xxx (though South might have bid 4H with that).

With the right defense this cannot be made whether you cover or duck the first trick.

Which is greater? $2^{128}$ or $3^{81}$

Which is greater?

$$2^{128}$$ or $$3^{81}$$

No calculators allowed.

Solution [Click here to expand/collapse]

$2^{128} = 340282366920938463463374607431768211456$

$3^{81} = 443426488243037769948249630619149892803$

So $3^{81} \gt 2^{128}$. QED (in about 4 hours).

Just kidding :)

Here is a proof without calculators.

Raise both sides to power of $\frac{1}{16}$.
We get $$2^8 \text{ vs } 3^{\frac{1}{16}}. 3^5$$ Now $3^{1/16} \gt e^{1/16} \gt 1 + \frac{1}{16}$ (using $e^x \gt 1 + x$)

Since $13 \times 16 = (15-2)(15+1) \lt 15^2 = 225 \lt 243$

We have that $1 + \frac{1}{16} \gt 1 + \frac{13}{243} = \frac{2^8}{3^5}$.
QED.

Classic trig equality

Show that

$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$



[Solution (math.stackexchange link).]

Yet another inequality

$x_1, x_2, \dots, x_n$ are $n$ positive real numbers with sum $S$, $n \gt 1$.

Show that

$$ \sum_{i=1}^{n} \frac{x_i}{S - x_i} \ge \frac{n}{n-1}$$


Solution [Click here to expand/collapse]

Using AM >= GM we have that

$\sum y_i \sum \frac{1}{y_i} \ge n^2$

Applying this to (wlog, assuming $S = 1$) $$\left(\sum_{i=1}^{n} \frac{1}{1-x_i}\right) \left(\sum_{i=1}^{n} (1 - x_i)\right) \ge n^2$$ i.e $$ \sum_{i=1}^{n} \frac{1}{1-x_i} \ge \frac{n^2}{n-1}$$ And so $$ \sum_{i=1}^{n} \left(\frac{1}{1-x_i} - 1\right) \ge \frac{n^2}{n-1} -n$$ Which gives us what we want to prove.

Splitting naturals into arithmetic progressions

The set of natural numbers $\{1,2,\dots\}$ is split into finite number of arithmetic progressions with common differences $d_1 \le d_2 \le \dots \le d_n$

Show that $$d_{n} = d_{n-1}$$

Eg: $\{4n+1\}, \{4n+3\}, \{2n\}$

Here $d_1 = 2, d_2 = d_3 = 4$.

Pointed nines

This is a hand from Mercercrest bridge club.

You are South. RHO opens 1D, you preempt 2S and opponents end up in 6H.

Partner leads the SK.

MPS Both
		Dummy ♠ 975 ♥ A9 ♦ AT542 ♣ KQ2
	You ♠ AQT862 ♥ 32 ♦ QJ3 ♣ 85

W	N	E	S
		1D	2S
3H	P	3S	P
4C	P	4H	P
4NT	P	5H	P
6H

3S was some kind of stopper asking bid.

Do you have a decision to make a trick one?


Looks like declarer has a singleton spade for the 4NT bid.

Declarer likely has 6+ hearts the DK and CA. If declarer has two or fewer diamonds, then declarer is cold.

So give declarer exactly 3 diamonds: a hand like ?, KQJxxx, K9x, Axx

Now if declarer has the singleton SJ, then you get squeezed in diamonds and spades, thanks to the diamond and spade nines!

So give partner the SJ.

Now if you play low, partner could potentially continue the SJ and now you are in charge of protecting spades and get squeezed again!

So you overtake with the A and declarer follows low. Now you play the SQ having partner keep the SJ.

Now when declarer draws trump throw your ST at the first opportunity! This will force partner to hold on to the SJ and the squeeze will not operate.

 

Mercercrest card reading hand

Mercercrest bridge club in Mercer island has its games every Tuesday evening and is considered one of the better fields in the Seattle area.

This is a hand from a recent game.

You are South and end up in 2S after the following auction

MPS None	Dummy ♠ xxx ♥ Tx ♦ KTx ♣ AJTxx
	You ♠ JT98x ♥ AJxx ♦ A9x ♣ x

W	N	E	S
1C	P	1H	1S
X*	2S	P	P
P

X was support, promising exactly 3 hearts. E/W play that 1D typically promises 4 and play 2/1 with strong NT (15-17).

LHO starts off with the Spade AKQ of trumps, RHO following twice and throwing a club on the third.

LHO then shifts to a low heart you play low from dummy and RHO plays low. Your spots are low enough that you have to win the J.

How will you play?



At the table I decided to play a club to the A and ruff a club back in hand. RHO followed with the Q.

The hand is almost an open book now.

LHO didn't open 1D, so cannot be 3=3=4=3, thus LHO has the CK. LHO also had the AKQ of spades. Since RHO played low on the heart, it is likely that RHO does not have both the K and Q.

LHO also has a balanced hand. If they had the HK they would have 15 points and thus would need both the DQ and DJ otherwise they would have opened 1NT. This would give RHO HQ and CQ.

So it is likely that LHO has the HQ and RHO has both the DQ and DJ (otherwise LHO would open 1NT).

Thus LHO has AKQ, Qxx, xxx, Kxxx and RHO had xx, Kxxx, QJxx, Qxx.


You can guarantee the contract by playing A and another heart.

If LHO wins and shifts to a diamond, thanks to your T and 9, you can capture RHO's Jack and then endplay them with a heart to lead a diamond back.

If RHO overtakes the HQ and cashes their 4th heart they are endplayed into leading a diamond back.

As it happened, LHO won the HQ and continued the CK!

Now you can ruff go to dummy with the DK and cash the good club to squeeze RHO for the overtrick. Made 9.

The four hands

MPS None	Dummy ♠ xxx ♥ Tx ♦ KTx ♣ AJTxx
West ♠ AKQ ♥ Qxx ♦ xxx ♣ Kxxx		East ♠ xx ♥ Kxxx ♦ QJxx ♣ Qxx
	You ♠ JT98x ♥ AJxx ♦ A9x ♣ x

W	N	E	S
1C	P	1H	1S
X*	2S	P	P
P

Extra chance?

This is a hand (with small changes) from the recent Bothell KO sectional (swiss teams).


You are South (and vul) and hold xx, KQT9xx, AKT, JT. RHO passes, 2H by partner which you pass and LHO balances with a double. Partner bids 3D (!), RHO bids 3S and you decide to bid 4H.

LHO leads a low spade and you see:

IMPS N/S	North ♠ Qx ♥ Axx ♦ 98xxxx ♣ Kx
	South ♠ xx ♥ KQT9xx ♦ AKT ♣ JT

W	N	E	S
			1H
P	2H	P	P
X	3D	3S	4H
P	P	P

RHO wins the SK and shifts to the DJ.

If you draw trumps they will be 2-2. How will you play?





Looks like the DJ is a singleton (LHO made a take out double) and RHO is angling for a ruff or two.

You have 9 tricks (assuming CA onside) and there seems to be no way to get the 10th trick, unless defense makes a mistake.

Is there a way to improve your chances of having the defense make a mistake?

Win the DJ cash HKQ drawing trumps and play the CJ!

If LHO ducks this (playing you for xx, KQxxxx, AK, JTX) then you are through. Play the CK and exit a club.

Either diamonds are 2-2, or the person who wins the last spade will be endplayed!

Escalator length

Two people go down an escalator (which is also moving down) taking one step at a time. The first person takes a step three times faster than the second. If the first person takes 75 steps total to reach the bottom and the second takes 50 steps, how many steps are showing in the escalator?

Removing perfect squares

Apparently this is from topcoder, but I believe there must be a more original source for this.

You start with $n$ cards numbered $1,2, \dots n$ placed in order along a line.

Now you make a pass through the cards and remove any that have a perfect square on them. Then you renumber the cards as $1,2 \dots K$ (making sure to maintain the ordering) and keep doing the process of removal and renumbering till there is only one card left.

What was the original number of that card? Can you give a formula in terms of $n$?

Surprising but easy

If

$$\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 1 $$

Show that

$$\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0 $$

Solution [Click here to expand/collapse]

Consider $$a + b + c = (a+b+c)\left(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}\right)$$
This is the same as $$ a + b + c + \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b}$$
And thus $$ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 0$$