Again this week we have a bunch of problems that constitute light snacks instead of a large banquet. (But don’t worry — there are McNuggets involved!) All of these problems hinge on the idea of constraints. You have to work with a limited set of numbers or objects with fixed denominations and combine them in various ways to produce a larger number or object with a specific value.

1.Use ALL and ONLY the numbers 1 1 9 9 with simple math (addition, subtraction, multiplication or division) to yield an answer of 10.

2.McDonald’s sells Chicken McNuggets in quantities of six pieces, nine pieces and 20 pieces. You can buy exactly 24 McNuggets by purchasing four orders of the six-piece size. It is not possible to buy exactly 23 McNuggets; nor is it possible to buy exactly 25 McNuggets. What is the largest whole number of which you cannot purchase an exact quantity of McNuggets?

3.What is the smallest number of common U.S. coins (pennies, nickels, dimes and quarters) that you need to have in your pocket to allow you to produce exact change for every amount from 1 cent to 99 cents?

Next: The challenge problem today is based on problem no. 3 above.

It concerns the planet of Noclinkia, which wanted to develop a monetary system similar to ours, with 100 cents to a dollar. Since their planet was short on metal, they scouted around space to see if there was a large stock of unused coins anywhere in the universe. They did indeed find a rich source, within the cushions, linings and upholstery of couches in a country called U.S.A. on planet Earth. So they used wormhole technology to suck large quantities of pennies, nickels, dimes and quarters from our couches to Noclinkia.

Now the Noclinkians, what with their chronic metal shortage and hypersensitivity to jingling change, are not happy with the denominations of our coins. “It requires slightly too many coins to make change for all amounts from 1 cent through 99,” they think. “Could we tinker with the denomination of one or more of these four coins so that it becomes possible to make change for all sub-dollar amounts with fewer coins?”

Can they? What is the smallest possible modification they can make to achieve this?

What if the Noclinkians got disgusted with all the lint, dirt, hair, gum, candy wrappers, sticky paper, smeared food residue, dried pet drool and yucky tarnish* on the coins they hoovered across space? What if they decided to melt all the coins and start anew with an entirely new currency system, not necessarily limited to four coin denominations? What advice would you give them then?

For math wizards, here’s an even tougher question. Suppose you knew that you needed to produce exact change ten times during the day, and each number between 1 and 99 was equally likely. What’s the smallest number of coins that you need to have to fulfill all requests on 95 percent of days or better, on average?

For today’s word challenge, consider the title of today’s puzzle. It intentionally plays on the similarity in sound between the words “constraint” and “restraint.” The substitution
of an incorrect word for a word with a similar sound, usually to comic effect, is a *malapropism*. Malapropisms have been a favorite ploy of public figures
from Stan Laurel to Yogi Berra. Here’s your task (malapropisms *italicized* and listed at the end for *charity*):

You have to come up with your own *virginal* malapropisms. You may quote one or two of your *favoring* malapropisms from external sources, but please exercise *constraint*. Of course, you may *remit* as many originals as you wish — no *statue* against that! I mean, no *constrictions* on that. Let’s make everyone *convalesce* with laughter!

(*clarity, original, favorite, restraint, submit, statute, restrictions, convulse*)

As usual, you can submit answers as comments here, and literary masterpieces and poems related to the challenges are welcome. The weekly prize, a copy of “Mathematical Puzzles of Sam Loyd” (edited by Martin Gardner), will go to someone who comes up with an especially interesting answer or who proposes a sufficiently intriguing puzzle for Lab readers to solve. (To submit a puzzle, send an e-mail message with “NEW LAB PUZZLE” in the subject line to tierneylab@nytimes.com. Please include the solution to the puzzle and indicate whether or not the puzzle is original.)

Question no. 1 today was submitted by Michael Pappas and question no. 2 by Casey Wills, who both get copies of the book. The first apparently was posed as an interview question at Bear Stearns, and the second was asked in a middle-school math competition. No pressure!

** Sofa Coin Gross-Out Challenge: What did I miss?*

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