A problem from the Leningrad Mathematical Olympiad: You have a set of 101 coins, and you know that it contains one counterfeit coin *X*. The 100 genuine coins all have the same weight, which is different from that of *X*. Using only two weighings in an equal-arm balance, how can you determine whether *X* is heavier or lighter than the genuine coins?

# Puzzles

# A Late Mystery

In Lloyd C. Douglas’ 1929 novel *Magnificent Obsession*, a doctor dies of a heart attack, leaving behind a journal written in cipher. The first page is shown here. Can you read it?

# Where Did Nigel Go?

A puzzle from the excellent Riddler feature at FiveThirtyEight, via Oliver Roeder’s 2018 collection *The Riddler*:

Your eccentric friend Nigel flies from Heathrow to an airport somewhere in the 48 contiguous states, then hires a car and drives around the country, touching the Atlantic and Pacific Oceans and the Gulf of Mexico, then returns to the airport at which he started and flies home. If he crossed the Ohio River once, the Missouri River twice, the Mississippi River three times, and the Continental Divide four times, then there’s one state that we can say for certain that he visited on his trip. What is it?

# Trainspotting

A puzzle from James F. Fixx’s *More Games for the Superintelligent*, 1976:

A man who likes trains walks occasionally to a nearby railroad track and waits for one to go by. Afterward he notes whether he saw a passenger train or a freight. After several years his notes show that 90 percent of the trains he’s seen have been passenger trains. One day he meets an official of the railroad and is surprised to learn that the passenger and freight trains on this line are precisely equal in number. If the man timed his trips to the track at random, why did he see such a disproportionate number of passenger trains?

# Triples

A brainteaser from the Soviet science magazine *Kvant*, via *Quantum*, January/February 1991:

Bobby found the sum of three consecutive integers, then of the next three consecutive integers, then multiplied these two sums together. Could the product have been 111,111,111?

# Black and White

By George E. Carpenter. White to mate in two moves.

# Line Work

Robert Bilinski proposed this problem in the April 2006 issue of *Crux Mathematicorum*. On square *ABCD*, two equilateral triangles are constructed, *ABE* internally and *BCF* externally, as shown. Prove that *D*, *E*, and *F* are collinear.

# A Century-Old Ghost

What does this mean?

PMVEB DWXZA XKKHQ RNFMJ VATAD YRJON FGRKD TSVWF TCRWC RLKRW ZCNBC FCONW FNOEZ QLEJB HUVLY OPFIN ZMHWC RZULG BGXLA GLZCZ GWXAH RITNW ZCQYR KFWVL CYGZE NQRNI JFEPS RWCZV TIZAQ LVEYI QVZMO RWQHL CBWZL HBPEF PROVE ZFWGZ RWLJG RANKZ ECVAW TRLBW URVSP KXWFR DOHAR RSRJJ NFJRT AXIJU RCRCP EVPGR ORAXA EFIQV QNIRV CNMTE LKHDC RXISG RGNLE RAFXO VBOBU CUXGT UEVBR ZSZSO RZIHE FVWCN OBPED ZGRAN IFIZD MFZEZ OVCJS DPRJH HVCRG IPCIF WHUKB NHKTV IVONS TNADX UNQDY PERRB PNSOR ZCLRE MLZKR YZNMN PJMQB RMJZL IKEFV CDRRN RHENC TKAXZ ESKDR GZCXD SQFGD CXSTE ZCZNI GFHGN ESUNR LYKDA AVAVX QYVEQ FMWET ZODJY RMLZJ QOBQ-

No one knows. Cryptologist Louis Kruh discovered it in the New York Public Library’s rare book room in 1993 among some old material from the U.S. Army Signal School. In 1915 first lieutenant Joseph O. Mauborgne had created what he believed was a more secure cipher than the ones currently in use, and had offered this challenge to see if his colleagues could break it. Kruh found no solution in the archive, and he published it in both *The Cryptogram* and *Cryptologia*, inviting their readers to try their hands at it. As far as I know, none succeeded.

Mauborgne described it as a “a simple, single-letter substitution cipher adapted to military use.” He invited the director of the Army Signal School to place it on a bulletin board and allow the officers there to work on it for three months and then to post the solution “to show why the standard method of attacking a substitution cipher fails in this case.” “If any attack upon this cipher is successful, I shall be glad to hear of it,” he wrote.

Kruh, who died in 2010, noted that “it was probably solved or otherwise deemed unsuitable for use because there is no knowledge of a new cipher being adopted by the Army around that time.” If a solution was found, I don’t believe anyone alive today knows what it is.

(Louis Kruh, “A 77-Year-Old Challenge Cipher,” *Cryptologia* 17:2 [April 1993], 172-174.)

# Setting the Date

A romantic puzzle from Albert H. Beiler’s *Recreations in the Theory of Numbers*:

An ardent swain said to his lady love, some years ago, ‘Once when a week ago last Tuesday was tomorrow, you said, “When a day just two fortnights hence will be yesterday, let us get married as it will be just this day next month.” Now sweetheart, we have waited just a fortnight so as it is now the second of the month let us figure out our wedding day.’

Beiler’s book came out in 1964, so he gives the answer Tuesday, March 17, 1936 — the couple are speaking on March 2, discussing a conversation they had on February 17. Obviously the answer is not unique — “Tuesday, March 17, 1908, is another solution but then the swain would not be very young.” Basically we need a leap year in which March 17 falls on a Tuesday. Beiler finds these occur also in 1964 and 1992 — and one did in 2020 as well.

# Black and White

By John Brown. White to mate in two moves.