Two challenges are offered as the tenth Coffeehouse problem, involving five separate items! Work each if you dare, or however many you wish to take up, as the year comes to a close. Have fun!

1. 4 CRYPTOGRAMS---As Charles and Maggie look back on the year's activities at Zeno's, they decided to construct four different cryptograms for the fine patrons who frequent the coffeehouse. Try to figure them out and see what's on their minds as the new year is at hand. Each contains a coded message, in which each letter is represented by the same code letter. Each cryptogram, however, has a different code! This type of puzzle shows the value of starting a problem with an incorrect assumption, and working through its consequences! Good luck!

First Cryptogram

XNV XHDV WZG HSPVEX HS WZGY ZQS KLHJHXHVE QHJJ YVXGYS K THSV CHPHCVSC.

Second Cryptogram

VCY VS WTY TLUEYOW WTGCAO WV MYLUC GO UYONYZW SVU WTVOY RTV EGOLAUYY RGWT XVQ.

Third Cryptogram

QCZLRKLUL CHLU BKL UFWPMCR...FP FLUCQCJ QIUFS WQ ELQBUCSWPN BKL CGCPL JFSLU.

Fourth Cryptogram

WFQZUE KUZQRC KDUQV: EZGZU YDGZ VF WDEO FNZQ VF WMKY CF VF PZN.

2. The GREEDY NEPHEWS---To celebrate the holiday season with his four young nephews, Charles set out a box in which he placed a number of dollar bills for them, and requested only that they each get to keep some of the contents. The first nephew grabbed up half of the bills, plus one more! The second took half of what remained, plus one additional bill, as did the third, and likewise the fourth nephew, who had thus emptied the box. Maggie asked Charles how many bills each nephew took, and how many he had placed in the box originally. Confused by the activity, he couldn't say----but Maggie took out a pencil and paper and figured it out!! As might you Zeno patrons who take up the challenge....

## From: Robert McWhorter

2. The GREEDY NEPHEWS--- 30 1st took 16 leaving 14 2nd took 8 leaving 6 3rd took 4 leaving 2 4th took 2 leaving box empty Reasoning: Start from bottom up. The 4th left the box empty after taking half of the remaining bills plus one. In math terms: 1/2x - 1 = 0 Letting x represent the number of bills in the box before the 4th empties the box. The number of bills at that time is 2. Then work up from there... From: James Parsly Cryptogram number 1 THE TIME YOU INVEST IN YOUR OWN ABILITIES WILL RETURN A FINE DIVIDEND. Cryptogram number 2 ONE OF THE HARDEST THINGS TO LEARN IS RESPECT FOR THOSE WHO DISAGREE WITH YOU. Cryptogram number 3 SOMEWHERE OVER THE RAINBOW...AN AEROSOL SPRAY IS DESTROYING THE OZONE LAYER. Cryptogram number 4 MODERN CREDIT CARDS: NEVER HAVE SO MANY OWED SO MUCH TO SO FEW. Greedy Nephews Worked it backwards assuming that the last nephew would only get 1 dollar: Last Nephew $1 Next to Last $3 Second $6 1st Nephew $12 Original Pot of Money $22 1st nephew gets 22/2 + 1 = 12 leaving 10 2nd nephew gets 10/5 + 1 = 6 leaving 4 3rd nephew gets 4/2 + 1 = 3 leaving 1 Last nephew gets 1 From: "James D. Egleston" Charles put thirty bills in the box. Since Nephew #4 took 1/2 of the bills left, plus one more, there must have been exactly two bills in the box when he got to it. For there to have been exactly two bills in the box when #4 got to it, #3 must have taken 4 of the remain six bills [4 = (6 x 1/2) + 1]. For there to have been six bills for #3 to get his greedy little hands on, #2 must have taken 8 of the remaining 14 bills [8 = (14 x 1/2) +1]. Finally, for there to have been 14 bills when #2 got to the box, #1 must have taken 16 of the 30 bills that Charles put in the box [16 = (30 x 1/2) + 1]. the whiz! From: DEACON The answer is 30 bills Happy New Year to you all!!!! From James_Dreier@brown.edu Wed Jan 8 12:50 EST 1997 Here are my solutions to Zeno's Tenth Challenge. THE CRYPTOGRAMS Cryptogram One The time you invest in your own abilities will return a fine dividend. Cryptogram Two One of the hardest things to learn is respect for those who disagree with you. Cryptogram Three Somewhere over the rainbow ... an aerosol spray is destroying the ozone layer. Cryptogram Four Moder credit cards: never have so many owed so much to so few. THE GREEDY NEPHEWS Charles put thirty dollar bills in the box. The first nephew took sixteen, leaving fourteen. The second nephew took eight, leaving six. The third nephew took four, leaving two. The fourth nephew took the remaining two. I especially liked the cryptograms, thanks! Jamie Dreier From: Lauri Kultti How many dollar bills uncle put in the hat? Let X denote the amount of bills for each nephew - in turn - before taking his share. Let nephews be - surprisingly - A,B,C and D. For D: D emptied the hat. So we are looking for X so that X - (1/2 x X + 1) = 0. Of course, X=2. For C: X - (1/2 x X +1) = 2 (the amount left to D). X=6. For B: X - (1/2 x X +1) = 6. X=14. For A: X - (1/2 x X +1) = 14. X=30. So, the amount of dollar bills in the hat at the beginning was 30. A took 16 (= 1/2 x 30 +1). B took 8 (= 1/2 x 14 +1). (There was 14 left after A took 16.) C took 4 (= 1/2 x 6 +1). (There was 6 left after A took 16 and B took 8.) D took the 2 bills left after other nephews have taken 16, 8 and 4. From: user #2. Original amt - $30 1st Nephew - $16 2nd " - $8 3rd " - $4 4th " - $2 atchley@vt.edu From: sruth there was $30 in dollar bills in the box.