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# Miscellaneous Puzzles on Mental Calculation

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#### A man married a widow, and they each already had children. Ten years later there was a pitched battle engaging the present family of twelve children. The mother ran to the father and cried, "Come at once! Your children and my children are fighting our children!" As the parents now had each nine children of their own, how many were born during the ten years?

Correct! Wrong!

Each parent had three children when they married, and six were born afterward.

#### What is the third lowest number that is both a triangular number and a square? Of course the numbers 1 and 36 are the two lowest that fulfill the conditions. What is the next number?

Correct! Wrong!

To find numbers that are both square and triangular, one has to solve the Pellian equation, 8 times a square plus 1 equals another square. The successive numbers for the first square are 1,6, 35, etc., and for the relative second squares 3, 17, 99, etc. Our answer is therefore 1,225 (352), which is both a square and a triangular number.

Correct! Wrong!

#### Colonel Crackham asked the junior members of his household at the breakfast table to write down five odd figures so that they will add up and make fourteen. Only one of them did it.

Correct! Wrong!

Write the following four numbers, composed of five odd figures, in the form of an addition sum, II, I, I, I, and they will add up to 14.

#### If we wanted to find a way of making the number 1,234,567 the difference between two squares, we could at once write down 617,284 and 617,283a half of the number plus'll and minus Ih respectively to be squared. But it will be found a little more difficult to discover two cubes the difference of which is 1,234,567.

Correct! Wrong!

The cube of 642 is 264,609,288, and the cube of 641 is 263,374,721, the difference being 1,234,567, as required

#### This is the kind of question that was very popular in Venice and elsewhere about the middle of the sixteenth century. Nicola Fontana, generally known as "Tartaglia" (the stammerer) was largely responsible for the invention. If a quarter of twenty is four, what would a third of ten be?

Correct! Wrong!

The answer must be 22/3. It is merely a sum in simple proportion: If 5 be 4, then 31/3 will be 22/3

#### When stopping at Mangleton-on-the-Bliss the Crackhams found the inhabitants of the town excited over some little local election. There were ten names of candidates on a proportional representation ballot. Voters should place No. I against the candidate of their first choice. They might also place No.2 against the candidate of their second choice, and so on until all the ten candidates have numbers placed against their names. The voters must mark their first choice, and any others may be marked or not as they wish. George proposed that they should discover in how many different ways the ballot might be marked by the voter

Correct! Wrong!

The number of different ways in which the ballot may be marked is 9,864,100

#### Can you find," Professor Rackbrane asked, "two consecutive cube numbers in integers whose difference shall be a square number? Thus the cube of 3 is 27, and the cube of 2 is 8, but the difference, 19, is not here a square number. What is the smallest possible case?"

Correct! Wrong!

The cube of 7 is 343, and the cube of 8 is 512; the difference, 169, is the square of 13

Correct! Wrong!

#### Can you find two three-figure square numbers (no zeros) that, when put together, will form a six-figure square number? Thus, 324 and 900 (the squares of 18 and 30) make 324,900, the square of 570, only there it happens there are two zeros. There is only one answer.

Correct! Wrong!

The number is 225,625 (the squares of 15 and 25), making the square of 475.