Dudeney, Amusements in Mathematics Problem 14 "This is queer, " said McCrank to his friend. "Twopence added to twopence is fourpence, and twopence multiplied by twopence is also fourpence." Of course, he was wrong in thinking you can multiply money by money. The multiplier must be regarded as an abstract number. It is true that two feet multiplied by two feet will make four square feet. Similarly, two pence multiplied by two pence will produce four square pence! And it will perplex the reader to say what a "square penny" is. But we will assume for the purposes of our puzzle that twopence multiplied by twopence is fourpence. Now, what two amounts of money will produce the next smallest possible result, the same in both cases, when added or multiplied in this manner? The two amounts need not be alike, but they must be those that can be paid in current coins of the realm.
Suppose both amounts are integers. Then ab = a+b implies b = a(b-1), so a must divide b. Similarly b must divide a. So they must be equal. Hence a 2 = 2a, so a = 0 or 2. Obviously 0 is disallowed by the wording of the question. So we do not get any solutions (apart from the 2.2 = 2+2 in the question). But the reason for all the talk about money is to show that the amounts do not have to be integers. They can be farthings or multiples of Â¼. So suppose the amounts are A farthings and B farthings. Then AB/16 = A/4 + B/4 or AB = 4A + 4B. Assume B >= A. We have B(A-4) = 4A, so B divides 4A. Hence B = 4A, 2A, 4A/3 or A. The first case gives A = 5, B = 20. The second case gives A = 6, B = 12. The third case gives A = 8, B = 8 and the last case gives A = 7, B = 28/3. The last case obviously fails. The third case is just 2d, 2d again, so we are left with the first and second case. We get the smallest result with A = 6, B = 12 or 1Â½d and 3d. Dudeney 1-25 (C) John Scholes jscholes@kalva.demon.co.uk 25 Oct 2002