Category Archives: Arithmetic

Six sixes


Digits and numbers are wonderful toys. The number of problems with them surpasses the imagination. Still, great minds think alike.
In 2013 I posed a classical problem to make as many numbers as possible by combining four fours.

In March I posed the challenge to make all the numbers of 0 to 20 by combining 5 5’s.

1) 6 6’s***/*****
Today’s problem is a sequel: make all the numbers 0 till 20 using 6 6’s. You can use brackets, +,-,*,/, ^and sqrt. You msy use ! for just one of the numbers.
You can check your solutions here and here

When googling, I found the puzzle masters at occupymath have posed the same challenge.

2) 7 7’s***/*****
Of course the problem can be generalized to 7 7’s. Make all numbers 0-24 using seven sevens, brackets, +-*/, no faculty.
You can check your solutions here and here

3) general ***/*****
The next problem is not, as you may have thought, to make all numbers 0-20 with 8 8’s. Rather the question is, can the numbers 0-20 with made with all digits? In base 11, in base 12?
For a start:
0 = (n-n)*(n+…+n)
1 = n/n + (n-n)*(n+…+n)
2 = n/n + n/n + (n-n)*(n+ … + n)
and so on.

Five five’s



Some of you may know the problem to make all the numbers from 1 to 20 with four fours.
I wrote about it here. It may come as a surprise to you that it is possible to make all the numbers from 0 to 100.

Today’s problem is to make all numbers from 0 to 20 using exactly five fives. For example:
(5*5 – 5*5) / 5 = 0

You can find my solutions here.

2020


It’s 2020, a new year! Here are a couple of 2020 related brainteasers:
1) 0-9!**/*****
Make each and every of the numbers 0-9 by combining the four digits 2, 0, 2, and 0 with arithmetic and mathematical operators. For example 0*(2+2)+0=0

2) two different digits**/*****
The year 2020 has only 2 different digits, 0 and 2. How many of those years do we have in this centur?

3) 0-9!**/*****
And as a sequel on 2): how many such years are there in this millennium?

You can check your solutions here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Dudeney’s Christmas puzzles


For years on end the famous British puzzle maker Henry Ernest Dudeney published puzzles in his weekly and monthly columns. Several of these have a Christmas theme, most of which I brought here together.

I know that some people claim other puzzles as Dudeneys Christmas puzzles, and I may or may not elaborate on that later. These have been selected because they have the word “Christmas” in them.

In his book “The Canterbury Puzzles” we find:
1) The Christmas Geese**/*****
Squire Hembrow, from Weston Zoyland—wherever that may be—proposed the following little arithmetical puzzle, from which it is probable that several somewhat similar modern ones have been derived: Farmer Rouse sent his man to market with a flock of geese, telling him that he might sell all or any of them, as he considered best, for he was sure the man knew how to make a good bargain. This is the report that Jabez made, though I have taken it out of the old Somerset dialect, which might puzzle some readers in a way not desired. “Well, first of all I sold Mr. Jasper Tyler half of the flock and half a goose over; then I sold Farmer Avent a third of what remained and a third of a goose over; then I sold Widow Foster a quarter of what remained and three-quarters of a goose over; and as I was coming home, whom should I meet but Ned Collier: so we had a mug of cider together at the Barley Mow, where I sold him exactly a fifth of what I had left, and gave him a fifth of a goose over for the missus. These nineteen that I have brought back I couldn’t get rid of at any price.” Now, how many geese did Farmer Rouse send to market? My humane readers may be relieved to know that no goose was divided or put to any inconvenience whatever by the sales.

You can check your solution here

2) Tasting the Plum Puddings**/*****

“Everybody, as I suppose, knows well that the number of different Christmas plum puddings that you taste will bring you the same number of lucky days in the new year. One of the guests (and his name has escaped my memory) brought with him a sheet of paper on which were drawn sixty-four puddings, and he said the puzzle was an allegory of a sort, and he intended to show how we might manage our pudding-tasting with as much dispatch as possible.” I fail to fully understand this fanciful and rather overstrained view of the puzzle. But it would appear that the puddings were arranged regularly, as I have shown them in the illustration, and that to strike out a pudding was to indicate that it had been duly tasted. You have simply to put the point of your pencil on the pudding in the top corner, bearing a sprig of holly, and strike out all the sixty-four puddings through their centres in twenty-one straight strokes. You can go up or down or horizontally, but not diagonally or obliquely; and you must never strike out a pudding twice, as that would imply a second and unnecessary tasting of those indigestible dainties. But the peculiar part of the thing is that you are required to taste the pudding that is seen steaming hot at the end of your tenth stroke, and to taste the one decked with holly in the bottom row the very last of all.

You can check your solution here

3) Under the Mistletoe Bough***/*****



“At the party was a widower who has but lately come into these parts,” says the record; “and, to be sure, he was an exceedingly melancholy man, for he did sit away from the company during the most part of the evening. We afterwards heard that he had been keeping a secret account of all the kisses that were given and received under the mistletoe bough. Truly, I would not have suffered any one to kiss me in that manner had I known that so unfair a watch was being kept. Other maids beside were in a like way shocked, as Betty Marchant has since told me.” But it seems that the melancholy widower was merely collecting material for the following little osculatory problem.

The company consisted of the Squire and his wife and six other married couples, one widower and three widows, twelve bachelors[Pg 92] and boys, and ten maidens and little girls. Now, everybody was found to have kissed everybody else, with the following exceptions and additions: No male, of course, kissed a male. No married man kissed a married woman, except his own wife. All the bachelors and boys kissed all the maidens and girls twice. The widower did not kiss anybody, and the widows did not kiss each other. The puzzle was to ascertain just how many kisses had been thus given under the mistletoe bough, assuming, as it is charitable to do, that every kiss was returned—the double act being counted as one kiss.

You can check your solution here

Dudeney lists another puzzle, “Buying presents”, but this involves now outdated British coins, for which reason I do not include it in this collection.

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Ikura


On January 18 I published some puzzles with the digits 1-9, each one to be used exactly once. In a publication by Denksport, the largest Dutch puzzle publisher, I found Ikura. Ikura in Japanese is the name of salmon caviar, and the first 20-50 hits in the duckduckgo search engine mostly referred to sushi and crosswords, so I guess Denksport made up this name themselves.

Below you find some examples of puzzles of this kind.

1) Ikura nr 1**/*****


2) Ikura nr 2**/*****

3) Ikura nr 3**/*****

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Ages


Ages (1)**/*****
“Did you know that this year the sum of our ages is a multiple of 8?” Jill asked John.
“Yes,” Tom answered. “And did you know that next year the product of our ages is a three digit number consisting of three times the same digit?”

Ages (2)**/*****
“What a coincidence,” Bill said, who had overheard their talk. “Next year the product of the ages of Bess and me will also be a three digit number consisting of three identical digits. But this year the sum of our ages is a two digit number consisting two identical digits.”

What are the ages of John, Jill, Bill and Bess?

You can check your solution here

Three squares


What number goes to the question mark?

1) Nr 1**/*****

You can check your solution here

2) Nr 2**/*****

You can check your solution here

3) Nr 3**/*****

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

2019


1) Make 2019 (1)**/*****
Take the series 1,2,3,4,5, etc. Insert + and – signs in between the numbers to make 2019. Do not use brackets. What is the shortest series?

2) Make 2019 (2)***/*****
Take the digits, 0-9. Keep them in exactly this ascending sequence and combine them wit +, -, *, /, ^, brackets and/or ! to make 2019.

3) Make 2019 (3)***/*****
Take the digits, 0-9. Order them in descending sequence (9, 8, 7, down till 0) and combine them with +, -, *, /, ^, brackets and/or ! to make 2019.

4) The digits 2, 0, 1, and 9***/*****
The year 2019 is made up of the digits 2,0,1 and 9. Combine them with all mathematical and arithmetic operators to make every number 0-20. For example: 0*1*2*9=0.

5) The digits 2, 0, 1, and 9***/*****
The 9 is the largest digit in “2019”. It is the square of the sum of the three other digits (9 = (2+0+1)^2). What is the next year when the largeest digit is euqal to the square of the other three digits?

You can check your solutions here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.